1
00:00:00,000 --> 00:00:00,920
2
00:00:00,920 --> 00:00:05,940
PROFESSOR: So we're slightly
into chapter seven, which is
3
00:00:05,940 --> 00:00:08,029
the algebra chapter.
4
00:00:08,029 --> 00:00:12,600
We're talking about a number of
algebraic objects, starting
5
00:00:12,600 --> 00:00:16,070
with integers and groups
and fields.
6
00:00:16,070 --> 00:00:16,970
Polynomials.
7
00:00:16,970 --> 00:00:21,700
Our objective in this chapter
is simply to get to finite
8
00:00:21,700 --> 00:00:27,370
fields so that you have some
sense what they are, how they
9
00:00:27,370 --> 00:00:30,780
can be constructed, what their
parameters are, how you can
10
00:00:30,780 --> 00:00:34,470
operate with them by addition,
multiplication, so forth, as
11
00:00:34,470 --> 00:00:36,630
you would expect in a field.
12
00:00:36,630 --> 00:00:39,440
Subtraction, division.
13
00:00:39,440 --> 00:00:42,380
And that's really all
we're aiming to do.
14
00:00:42,380 --> 00:00:46,750
I'm trying to give you a short
course in algebra, really, in
15
00:00:46,750 --> 00:00:49,200
two lectures or fewer.
16
00:00:49,200 --> 00:00:53,100
And clearly I'm going to
miss a lot of things.
17
00:00:53,100 --> 00:00:55,155
In particular, I'm not going
to cover even everything
18
00:00:55,155 --> 00:00:58,830
that's in chapter seven, which
itself is a highly compressed
19
00:00:58,830 --> 00:01:01,950
introduction to finite fields.
20
00:01:01,950 --> 00:01:05,660
I'm trying to do this while
remaining faithful to the
21
00:01:05,660 --> 00:01:09,020
philosophy of this course and
other courses at MIT, which is
22
00:01:09,020 --> 00:01:12,880
that you should really prove
everything and show why things
23
00:01:12,880 --> 00:01:15,810
are true, and not simply
make assertions.
24
00:01:15,810 --> 00:01:18,970
So it's a little tough and it
forces me to go a little fast,
25
00:01:18,970 --> 00:01:22,820
but I hope that you can keep
up, and especially with the
26
00:01:22,820 --> 00:01:26,680
assistance of the notes or the
many possible other things you
27
00:01:26,680 --> 00:01:29,680
could read on this subject,
which are listed in the notes,
28
00:01:29,680 --> 00:01:31,390
you'll be able to keep up.
29
00:01:31,390 --> 00:01:33,960
And some of you, of course, have
seen this in other places
30
00:01:33,960 --> 00:01:37,490
in more extended form.
31
00:01:37,490 --> 00:01:41,340
Now, this will get us in a
position to start to talk
32
00:01:41,340 --> 00:01:46,150
about Reed-Solomon codes, which
are the single major
33
00:01:46,150 --> 00:01:50,250
accomplishment of the field of
algebraic coding theory.
34
00:01:50,250 --> 00:01:53,920
Certainly for getting to
capacity on the additive white
35
00:01:53,920 --> 00:01:58,835
Gaussian noise channel and for
lots of other things, they're
36
00:01:58,835 --> 00:02:04,340
an extremely useful and widely
implemented class of codes.
37
00:02:04,340 --> 00:02:06,790
And we'll be able to maybe just
get to the beginning of
38
00:02:06,790 --> 00:02:08,910
that by Wednesday.
39
00:02:08,910 --> 00:02:11,610
And then I won't be here again
next week, but fortunately we
40
00:02:11,610 --> 00:02:17,760
have an expert on campus who is
far more expert than I in
41
00:02:17,760 --> 00:02:21,180
Reed-Solomon codes, their
decoding algorithms, who has
42
00:02:21,180 --> 00:02:25,360
agreed to talk for two lectures,
maybe one more
43
00:02:25,360 --> 00:02:29,570
focused on Reed-Solomon codes
and one, I hope, on his whole
44
00:02:29,570 --> 00:02:31,040
philosophy of life.
45
00:02:31,040 --> 00:02:31,580
Perhaps.
46
00:02:31,580 --> 00:02:33,830
I don't know how it's going to
come out, but I'll see it on
47
00:02:33,830 --> 00:02:36,710
TV when I get back.
48
00:02:36,710 --> 00:02:40,740
Anyway, Ralf Koetter will be
lecturer for Monday and
49
00:02:40,740 --> 00:02:42,900
Wednesday next week.
50
00:02:42,900 --> 00:02:46,880
I think you'll enjoy
the change of pace.
51
00:02:46,880 --> 00:02:47,360
OK.
52
00:02:47,360 --> 00:02:49,780
So where are we in
chapter seven?
53
00:02:49,780 --> 00:02:50,800
We're not very far.
54
00:02:50,800 --> 00:02:54,250
We're talking about these
various algebraic objects.
55
00:02:54,250 --> 00:02:59,030
We've started with integers
just to get you
56
00:02:59,030 --> 00:02:59,860
into the feel of it.
57
00:02:59,860 --> 00:03:04,260
We mainly talked about integer
factorization, the Euclidean
58
00:03:04,260 --> 00:03:06,860
division algorithm, things that
you've known for a very
59
00:03:06,860 --> 00:03:11,640
long time, basically here
because A, we're going to be
60
00:03:11,640 --> 00:03:15,100
using integers as we go along,
and their factorization
61
00:03:15,100 --> 00:03:18,360
properties, B, it's a model for
polynomials, which behave
62
00:03:18,360 --> 00:03:22,080
very much the same way as
integers because they're both
63
00:03:22,080 --> 00:03:23,390
principal ideal domains.
64
00:03:23,390 --> 00:03:28,380
In particular, we looked at the
integers mod n with the
65
00:03:28,380 --> 00:03:34,900
rules of mod n arithmetic, which
we're going to call Zn.
66
00:03:34,900 --> 00:03:38,800
This is simply 0 through n
minus 1 with the mod n
67
00:03:38,800 --> 00:03:41,440
arithmetic rules.
68
00:03:41,440 --> 00:03:44,920
And then we went on to groups.
69
00:03:44,920 --> 00:03:49,140
We first gave the standard
axioms for groups, and then I
70
00:03:49,140 --> 00:03:53,000
gave you an alternative set of
axioms which focused on this
71
00:03:53,000 --> 00:03:54,370
permutation property.
72
00:03:54,370 --> 00:03:59,590
If you add, I'm calling the
group operation addition,
73
00:03:59,590 --> 00:04:03,450
because essentially all the
groups we talk about are going
74
00:04:03,450 --> 00:04:05,240
to be abelian --
75
00:04:05,240 --> 00:04:08,950
if we add a group element
to the group,
76
00:04:08,950 --> 00:04:10,530
what do we get back?
77
00:04:10,530 --> 00:04:12,800
We get the whole group again.
78
00:04:12,800 --> 00:04:15,040
It's permuted, it's the
entire group, it's
79
00:04:15,040 --> 00:04:16,589
in a different order.
80
00:04:16,589 --> 00:04:18,050
All right?
81
00:04:18,050 --> 00:04:21,610
And with this and the identity,
this plus the
82
00:04:21,610 --> 00:04:26,770
identity and the associativity
axiom are also a sufficient
83
00:04:26,770 --> 00:04:28,960
set of axioms for the group.
84
00:04:28,960 --> 00:04:32,010
And I think this is really the
most useful thing to think
85
00:04:32,010 --> 00:04:33,540
about with a group.
86
00:04:33,540 --> 00:04:35,460
We've also called it the
group property when we
87
00:04:35,460 --> 00:04:36,845
talked about codes.
88
00:04:36,845 --> 00:04:40,000
You know, if you add the code
word to all the elements in
89
00:04:40,000 --> 00:04:41,440
the code, you get
the code back.
90
00:04:41,440 --> 00:04:44,190
91
00:04:44,190 --> 00:04:47,090
And you saw how useful that
was for seeing certain
92
00:04:47,090 --> 00:04:51,110
symmetry properties of minor
linear block codes.
93
00:04:51,110 --> 00:04:54,150
So we even talked about cyclic
groups, specifically finite
94
00:04:54,150 --> 00:04:55,570
cyclic groups.
95
00:04:55,570 --> 00:04:59,340
And we showed that all
them basically are
96
00:04:59,340 --> 00:05:01,600
isomorphic to z mod n.
97
00:05:01,600 --> 00:05:04,305
98
00:05:04,305 --> 00:05:06,800
A cyclic group is defined
by a single generator.
99
00:05:06,800 --> 00:05:10,830
If we identify that generator
with one, G plus G is
100
00:05:10,830 --> 00:05:14,070
identified with 2, and so forth,
then we get an addition
101
00:05:14,070 --> 00:05:16,840
table which is exactly the same
addition table as Zn.
102
00:05:16,840 --> 00:05:19,770
And that's what we mean when
we say two groups are
103
00:05:19,770 --> 00:05:20,480
isomorphic.
104
00:05:20,480 --> 00:05:23,680
So all finite cyclic groups
look like Zn.
105
00:05:23,680 --> 00:05:26,900
You can think of them as
being images of Zn.
106
00:05:26,900 --> 00:05:28,400
All right?
107
00:05:28,400 --> 00:05:30,930
It's the only one you
need to know about.
108
00:05:30,930 --> 00:05:31,500
OK.
109
00:05:31,500 --> 00:05:37,950
So that's where we are any
questions on this material?
110
00:05:37,950 --> 00:05:41,110
Pretty easy, I think.
111
00:05:41,110 --> 00:05:43,310
Terribly easy if you've ever
seen any of this before.
112
00:05:43,310 --> 00:05:48,080
Probably takes a little
absorbing if you haven't.
113
00:05:48,080 --> 00:05:48,330
OK.
114
00:05:48,330 --> 00:05:53,880
Now the next natural subject
to talk about is subgroups.
115
00:05:53,880 --> 00:05:55,950
And what is a subgroup?
116
00:05:55,950 --> 00:06:00,640
A subgroup simply a subset of
elements in the group which,
117
00:06:00,640 --> 00:06:04,680
together with the group
operation already specified in
118
00:06:04,680 --> 00:06:10,345
G, which we're calling circle
plus, is itself a group.
119
00:06:10,345 --> 00:06:12,870
120
00:06:12,870 --> 00:06:13,810
What does that mean?
121
00:06:13,810 --> 00:06:17,220
Well, associativity comes for
free, because we already have
122
00:06:17,220 --> 00:06:21,840
that property for circle plus
in G. Obviously H has to
123
00:06:21,840 --> 00:06:28,460
include the identity in order
to satisfy the group axioms.
124
00:06:28,460 --> 00:06:31,290
And finally, the third group
axiom is this permutation or
125
00:06:31,290 --> 00:06:34,770
group property that if we add
any element of the subgroup to
126
00:06:34,770 --> 00:06:38,430
itself, we have to stay within
the subgroup and ultimately
127
00:06:38,430 --> 00:06:40,270
generate the whole subgroup.
128
00:06:40,270 --> 00:06:43,150
That means that subtraction,
129
00:06:43,150 --> 00:06:46,990
cancellation hold in the subgroup.
130
00:06:46,990 --> 00:06:48,310
All right?
131
00:06:48,310 --> 00:06:51,390
So that's clear.
132
00:06:51,390 --> 00:06:55,460
What's an example
of a subgroup?
133
00:06:55,460 --> 00:06:57,180
If we have --
134
00:06:57,180 --> 00:07:01,180
we talked about Z10
as the group.
135
00:07:01,180 --> 00:07:02,310
What would be a subgroup?
136
00:07:02,310 --> 00:07:03,560
Anybody?
137
00:07:03,560 --> 00:07:08,167
138
00:07:08,167 --> 00:07:10,120
AUDIENCE: [INAUDIBLE]
139
00:07:10,120 --> 00:07:12,220
PROFESSOR: 0 to 4.
140
00:07:12,220 --> 00:07:12,750
OK.
141
00:07:12,750 --> 00:07:18,970
So you're proposing that H is
the elements 0, 1, 2, 3, 4 out
142
00:07:18,970 --> 00:07:22,730
of G. Does that work?
143
00:07:22,730 --> 00:07:25,670
This doesn't include 0.
144
00:07:25,670 --> 00:07:27,110
But suppose I add 3 and 4?
145
00:07:27,110 --> 00:07:30,100
146
00:07:30,100 --> 00:07:34,950
AUDIENCE: I assume that you
have modules for a reason.
147
00:07:34,950 --> 00:07:35,650
PROFESSOR: No.
148
00:07:35,650 --> 00:07:41,370
In this group, the group
operations modulo 10 addition.
149
00:07:41,370 --> 00:07:43,720
Subgroup has to have the
same operation as the
150
00:07:43,720 --> 00:07:44,970
group it came from.
151
00:07:44,970 --> 00:07:48,600
152
00:07:48,600 --> 00:07:53,840
What you've got here is you've
already got, in essence, a
153
00:07:53,840 --> 00:07:57,036
quotient group.
154
00:07:57,036 --> 00:07:58,630
Or at least that's where
you're headed.
155
00:07:58,630 --> 00:08:01,130
So this is not a subgroup.
156
00:08:01,130 --> 00:08:02,300
Thank you for the suggestion.
157
00:08:02,300 --> 00:08:03,310
Fails.
158
00:08:03,310 --> 00:08:03,970
Anyone else?
159
00:08:03,970 --> 00:08:04,300
What?
160
00:08:04,300 --> 00:08:05,250
AUDIENCE: 0,1?
161
00:08:05,250 --> 00:08:07,020
PROFESSOR: 0,1.
162
00:08:07,020 --> 00:08:07,230
OK.
163
00:08:07,230 --> 00:08:10,610
Let's try that.
164
00:08:10,610 --> 00:08:13,990
And it contains the identity
0 plus 1 is
165
00:08:13,990 --> 00:08:14,700
certainly in the group.
166
00:08:14,700 --> 00:08:17,160
How about 1 plus 1?
167
00:08:17,160 --> 00:08:20,810
It's not in the group.
168
00:08:20,810 --> 00:08:21,617
0 and 5?
169
00:08:21,617 --> 00:08:22,867
That sounds more promising.
170
00:08:22,867 --> 00:08:28,910
171
00:08:28,910 --> 00:08:31,580
Now, 0 plus -- what's the
addition table of this?
172
00:08:31,580 --> 00:08:34,419
173
00:08:34,419 --> 00:08:35,669
0, 5, 0, 5, 0, 5.
174
00:08:35,669 --> 00:08:38,780
175
00:08:38,780 --> 00:08:39,940
What's 5 plus 5?
176
00:08:39,940 --> 00:08:40,570
It's 10.
177
00:08:40,570 --> 00:08:43,360
But mod-10 , that's 0.
178
00:08:43,360 --> 00:08:45,290
So it seems we do
have a group.
179
00:08:45,290 --> 00:08:51,750
And in fact, this is a finite
cyclic group generated by 5,
180
00:08:51,750 --> 00:08:57,500
and has two elements, so
it's isomorphic to Z2.
181
00:08:57,500 --> 00:08:59,740
In other words, the addition
table looks just like the
182
00:08:59,740 --> 00:09:01,600
addition table of Z2
with a relabeling.
183
00:09:01,600 --> 00:09:05,500
184
00:09:05,500 --> 00:09:06,860
OK.
185
00:09:06,860 --> 00:09:10,080
Any other subgroups of Z10?
186
00:09:10,080 --> 00:09:11,280
The even integers.
187
00:09:11,280 --> 00:09:11,680
There.
188
00:09:11,680 --> 00:09:14,010
Now we're really smoking.
189
00:09:14,010 --> 00:09:18,890
H equals 0,2,4,6,8.
190
00:09:18,890 --> 00:09:20,580
Those are all the even
integers in Z10.
191
00:09:20,580 --> 00:09:23,330
192
00:09:23,330 --> 00:09:26,380
And again, evens plus
evens equal evens.
193
00:09:26,380 --> 00:09:31,880
So we get a group of five
elements, satisfies is the
194
00:09:31,880 --> 00:09:39,320
group property, and it's
isomorphic to what?
195
00:09:39,320 --> 00:09:39,845
Z5 --
196
00:09:39,845 --> 00:09:41,310
yeah.
197
00:09:41,310 --> 00:09:44,490
This is obviously the same group
is 0, 1, 2, 3, 4, just
198
00:09:44,490 --> 00:09:45,740
doubling everything.
199
00:09:45,740 --> 00:09:48,700
200
00:09:48,700 --> 00:09:49,720
Operates the same way.
201
00:09:49,720 --> 00:09:50,970
So it's change of labels.
202
00:09:50,970 --> 00:09:53,350
203
00:09:53,350 --> 00:09:53,460
OK.
204
00:09:53,460 --> 00:09:55,485
So there are some examples
of subgroups.
205
00:09:55,485 --> 00:09:59,030
206
00:09:59,030 --> 00:10:03,700
Let's take another
good example.
207
00:10:03,700 --> 00:10:05,680
Let's just take the set
of all integers.
208
00:10:05,680 --> 00:10:07,780
That's an infinite group.
209
00:10:07,780 --> 00:10:12,400
Mathematicians call it cyclic,
even though it doesn't cycle.
210
00:10:12,400 --> 00:10:13,800
What's a subgroup of that?
211
00:10:13,800 --> 00:10:23,530
212
00:10:23,530 --> 00:10:24,460
AUDIENCE: Even integers.
213
00:10:24,460 --> 00:10:25,450
PROFESSOR: All even integers!
214
00:10:25,450 --> 00:10:28,140
Very good.
215
00:10:28,140 --> 00:10:28,420
Check.
216
00:10:28,420 --> 00:10:29,670
Does that include 0?
217
00:10:29,670 --> 00:10:30,370
Yes.
218
00:10:30,370 --> 00:10:31,910
Does it have the group
property, even
219
00:10:31,910 --> 00:10:34,240
plus even is even?
220
00:10:34,240 --> 00:10:36,670
Clearly subtraction holds,
and so forth.
221
00:10:36,670 --> 00:10:39,560
So this is the subgroup.
222
00:10:39,560 --> 00:10:42,190
223
00:10:42,190 --> 00:10:44,860
Interestingly, there's a
one-to-one correspondence
224
00:10:44,860 --> 00:10:47,440
between Z and 2Z, so the
subgroup is as big as the
225
00:10:47,440 --> 00:10:50,420
group itself.
226
00:10:50,420 --> 00:10:56,940
You get into the whole issue
of transfinite numbers.
227
00:10:56,940 --> 00:10:58,570
But that's not where
we're headed here.
228
00:10:58,570 --> 00:11:01,150
229
00:11:01,150 --> 00:11:03,030
OK.
230
00:11:03,030 --> 00:11:05,390
We'll just keep that
in mind for now.
231
00:11:05,390 --> 00:11:13,350
Obviously 3Z, 4Z, 5Z, and all
the multiples of n are going
232
00:11:13,350 --> 00:11:17,390
to be a subgroup of the integers
for any integer n.
233
00:11:17,390 --> 00:11:21,930
So more generally, we could
take H equals nZ.
234
00:11:21,930 --> 00:11:34,400
235
00:11:34,400 --> 00:11:35,650
What's a coset?
236
00:11:35,650 --> 00:11:38,100
237
00:11:38,100 --> 00:11:42,450
Coset is also, in the abelian
case, called a
238
00:11:42,450 --> 00:11:45,306
translate of a subgroup.
239
00:11:45,306 --> 00:11:50,820
240
00:11:50,820 --> 00:11:59,820
A coset, for instance, is in the
form H plus G for some G
241
00:11:59,820 --> 00:12:01,070
in the group, not
in the subgroup.
242
00:12:01,070 --> 00:12:04,670
243
00:12:04,670 --> 00:12:09,730
Now if G is in the subgroup,
we get nothing.
244
00:12:09,730 --> 00:12:16,970
The coset is just H again by
the group property of H. So
245
00:12:16,970 --> 00:12:25,700
the interesting cases are where
G is not in H. Let me
246
00:12:25,700 --> 00:12:27,820
give you some examples.
247
00:12:27,820 --> 00:12:41,290
Let's take G equals Z10 and
H equals 0,2,4,6,8.
248
00:12:41,290 --> 00:12:47,720
You might call that 2Z10 It's
the set of all elements which
249
00:12:47,720 --> 00:12:52,050
are twice the elements in Z10.
250
00:12:52,050 --> 00:12:52,500
OK.
251
00:12:52,500 --> 00:12:54,600
What is a coset?
252
00:12:54,600 --> 00:12:58,180
If I add any element in
this group to H --
253
00:12:58,180 --> 00:13:01,780
let's add one, for instance.
254
00:13:01,780 --> 00:13:13,585
So H plus 1 consists of the
elements 1,3,5,7,9.
255
00:13:13,585 --> 00:13:14,835
Well, that's interesting.
256
00:13:14,835 --> 00:13:17,000
257
00:13:17,000 --> 00:13:25,130
That seems to exhaust all
of the elements of Z10.
258
00:13:25,130 --> 00:13:33,370
Let's take the Z equals the Z10,
and H simply equal to 0
259
00:13:33,370 --> 00:13:35,550
and 5, which we might
call 5Z10.
260
00:13:35,550 --> 00:13:39,650
261
00:13:39,650 --> 00:13:42,000
And all right.
262
00:13:42,000 --> 00:13:46,570
Now H plus 0, H plus 5 is
just equal to itself.
263
00:13:46,570 --> 00:13:50,650
H plus 1 is equal to 1,6.
264
00:13:50,650 --> 00:13:55,240
H plus 2 is equal to 2,7.
265
00:13:55,240 --> 00:14:01,040
H plus 3 is equal to 3, 8.
266
00:14:01,040 --> 00:14:10,770
H plus 4 is equal to 4,9.
267
00:14:10,770 --> 00:14:18,320
So we begin to see some
properties of cosets here for
268
00:14:18,320 --> 00:14:22,050
which proofs are given
in the notes.
269
00:14:22,050 --> 00:14:27,760
First of all is that
two cosets are
270
00:14:27,760 --> 00:14:31,020
either the same or disjoint.
271
00:14:31,020 --> 00:14:36,760
H plus 2 is the same as H. H
plus 1 is completely disjoint
272
00:14:36,760 --> 00:14:42,520
from H. Same over here.
273
00:14:42,520 --> 00:14:49,450
If I had H plus 5, that would
be the same as H. H plus 6
274
00:14:49,450 --> 00:14:52,360
would be completely disjoint
from H and would be the same
275
00:14:52,360 --> 00:14:54,300
as H plus 1.
276
00:14:54,300 --> 00:14:58,650
In fact, I can take any of the
elements of a coset as its
277
00:14:58,650 --> 00:15:01,630
representative, and I'm going
to get the same coset, take
278
00:15:01,630 --> 00:15:04,040
any element outside the coset,
and I'll get a completely
279
00:15:04,040 --> 00:15:05,010
distinct coset.
280
00:15:05,010 --> 00:15:07,370
This follows just very
easily from the
281
00:15:07,370 --> 00:15:10,610
cancellation property.
282
00:15:10,610 --> 00:15:11,900
All right?
283
00:15:11,900 --> 00:15:24,170
So the cosets, the distinct
cosets, form a disjoint
284
00:15:24,170 --> 00:15:36,780
partition of G. We certainly
have a coset that contains
285
00:15:36,780 --> 00:15:40,210
every element of G. Just take H
plus that element G. That's
286
00:15:40,210 --> 00:15:43,870
going to contain G because
H contains 0.
287
00:15:43,870 --> 00:15:49,990
So there is a coset that
contains every element of G.
288
00:15:49,990 --> 00:15:54,030
Any two cosets, distinct cosets,
are disjoint, so
289
00:15:54,030 --> 00:15:56,350
that's what we mean by
a disjoint partition.
290
00:15:56,350 --> 00:16:00,470
We list all the elements
of G in this way.
291
00:16:00,470 --> 00:16:03,130
292
00:16:03,130 --> 00:16:09,700
And in the finite case, that
gives us a early famous
293
00:16:09,700 --> 00:16:14,220
theorem attributed
to Lagrange.
294
00:16:14,220 --> 00:16:17,440
295
00:16:17,440 --> 00:16:18,600
What does that mean?
296
00:16:18,600 --> 00:16:35,800
This means that H has to divide
G. If G has size 1,10
297
00:16:35,800 --> 00:16:40,970
and all the cosets have the same
size, by the way, again
298
00:16:40,970 --> 00:16:43,450
by the cancellation
property --
299
00:16:43,450 --> 00:16:48,040
this means that G has to consist
of an integer number
300
00:16:48,040 --> 00:16:53,410
of cosets, all of which have
the size of H. OK?
301
00:16:53,410 --> 00:16:57,410
And therefore some integer times
H is equal to H, which
302
00:16:57,410 --> 00:17:01,240
is the same thing as
H divides G. OK?
303
00:17:01,240 --> 00:17:13,170
304
00:17:13,170 --> 00:17:18,210
If G is a finite group, I mean
by this kind of determinant
305
00:17:18,210 --> 00:17:22,619
notation, the size of H. The
size of H divides the size of
306
00:17:22,619 --> 00:17:26,890
G. Or more elegantly, the
cardinality of H divides the
307
00:17:26,890 --> 00:17:29,350
cardinality of G. But
why say cardinality
308
00:17:29,350 --> 00:17:32,540
when you can say size?
309
00:17:32,540 --> 00:17:33,792
I shouldn't have put it here.
310
00:17:33,792 --> 00:17:36,990
311
00:17:36,990 --> 00:17:41,210
Where H is any subgroup, any
subgroup, of course, that's
312
00:17:41,210 --> 00:17:44,850
finite is itself
a finite group.
313
00:17:44,850 --> 00:17:47,630
So that's going to turn out to
be quite a powerful theorem.
314
00:17:47,630 --> 00:17:51,770
And it just follows this
little exercise.
315
00:17:51,770 --> 00:17:54,500
And we see it's satisfied,
certainly,
316
00:17:54,500 --> 00:17:56,680
by these two examples.
317
00:17:56,680 --> 00:17:59,270
318
00:17:59,270 --> 00:18:06,970
In fact, it's pretty easy to see
that the subgroups of Zm
319
00:18:06,970 --> 00:18:10,720
are going to correspond to the
divisors of m in the same way.
320
00:18:10,720 --> 00:18:14,050
We're going to get a subgroup
for every divisor of m, and in
321
00:18:14,050 --> 00:18:16,375
the case of cyclic groups,
this is the way it's
322
00:18:16,375 --> 00:18:17,310
going to come out.
323
00:18:17,310 --> 00:18:18,800
Just pick any divisor.
324
00:18:18,800 --> 00:18:24,150
You get a subgroup
isomorphic to Zd.
325
00:18:24,150 --> 00:18:27,970
Again, this is done with
more care in the notes.
326
00:18:27,970 --> 00:18:30,100
Suppose we have the infinite
case here.
327
00:18:30,100 --> 00:18:32,920
Suppose we have Z and 2Z.
328
00:18:32,920 --> 00:18:36,750
So let me draw that case.
329
00:18:36,750 --> 00:18:41,150
G equals Z. H equals 2Z.
330
00:18:41,150 --> 00:18:43,680
331
00:18:43,680 --> 00:18:44,970
And all right.
332
00:18:44,970 --> 00:18:48,400
What's H?
333
00:18:48,400 --> 00:18:58,730
H is the set dot dot dot minus
2,0,2,4, dot dot dot.
334
00:18:58,730 --> 00:19:06,870
So this is H. And what's H plus
1, or in fact, plus any
335
00:19:06,870 --> 00:19:08,520
odd number?
336
00:19:08,520 --> 00:19:11,210
It's going to be the
odd integers.
337
00:19:11,210 --> 00:19:16,120
H equals minus 1,1,3,5,
and so forth.
338
00:19:16,120 --> 00:19:18,790
339
00:19:18,790 --> 00:19:26,100
So again, this works in the
infinite case, that the
340
00:19:26,100 --> 00:19:29,850
distinct cosets form a disjoint
partition of an
341
00:19:29,850 --> 00:19:33,105
infinite group G. But of
course, we don't get
342
00:19:33,105 --> 00:19:38,040
Lagrange's theorem as a
corollary, because I already
343
00:19:38,040 --> 00:19:40,250
said, there's a one-to-one
correspondence between Z and
344
00:19:40,250 --> 00:19:44,360
2Z, paradoxically.
345
00:19:44,360 --> 00:19:48,160
So the cardinality of Z
and 2Z are the same.
346
00:19:48,160 --> 00:19:50,130
More elegant language.
347
00:19:50,130 --> 00:19:53,070
Nonetheless, you see, this
is a useful partition and
348
00:19:53,070 --> 00:19:56,220
standard partition
into the even
349
00:19:56,220 --> 00:19:59,030
integers and the odd integers.
350
00:19:59,030 --> 00:20:04,990
And we could also write this as
2Z and this is 2Z plus 1.
351
00:20:04,990 --> 00:20:06,930
So we can divide the
integers into even
352
00:20:06,930 --> 00:20:08,180
integers and odd integers.
353
00:20:08,180 --> 00:20:10,640
354
00:20:10,640 --> 00:20:11,350
All right.
355
00:20:11,350 --> 00:20:18,360
Now we can actually add cosets,
subtract cosets.
356
00:20:18,360 --> 00:20:21,110
In these cases, we can
even multiply cosets.
357
00:20:21,110 --> 00:20:23,090
But let's just talk
about staying
358
00:20:23,090 --> 00:20:25,760
within the group operation.
359
00:20:25,760 --> 00:20:27,600
Given any abelian --
360
00:20:27,600 --> 00:20:29,870
let's continue to say
G is abelian --
361
00:20:29,870 --> 00:20:31,670
how would you add two cosets?
362
00:20:31,670 --> 00:20:34,340
363
00:20:34,340 --> 00:20:42,060
Coset addition is defined
as follows.
364
00:20:42,060 --> 00:20:44,730
H plus G --
365
00:20:44,730 --> 00:20:48,290
we want to have some addition
operation, which I'll just
366
00:20:48,290 --> 00:20:50,860
indicate by plus --
367
00:20:50,860 --> 00:20:54,810
H plus G prime, what's
going to equal?
368
00:20:54,810 --> 00:21:04,065
We define that to equal
H plus G plus G prime.
369
00:21:04,065 --> 00:21:07,490
370
00:21:07,490 --> 00:21:09,110
And that makes sense.
371
00:21:09,110 --> 00:21:15,830
I mean, if we really write all
this out, we get H plus H plus
372
00:21:15,830 --> 00:21:18,940
G plus G prime.
373
00:21:18,940 --> 00:21:21,490
H plus H is just H again.
374
00:21:21,490 --> 00:21:24,740
So it's sort of a
proof of that.
375
00:21:24,740 --> 00:21:29,460
If you go through in detail, any
element of this coset plus
376
00:21:29,460 --> 00:21:32,330
any element of this coset
is going to be an
377
00:21:32,330 --> 00:21:34,930
element of this coset.
378
00:21:34,930 --> 00:21:40,380
So this itself is a coset.
379
00:21:40,380 --> 00:21:42,680
So we now have an addition
table for cosets.
380
00:21:42,680 --> 00:21:47,780
381
00:21:47,780 --> 00:21:51,340
So in fact, it's easy to show
that the cosets themselves
382
00:21:51,340 --> 00:21:56,700
form a group called
a quotient group.
383
00:21:56,700 --> 00:21:57,950
Start over here.
384
00:21:57,950 --> 00:22:05,400
385
00:22:05,400 --> 00:22:16,073
Cosets of H in G under
coset addition --
386
00:22:16,073 --> 00:22:17,323
that's going to be Z --
387
00:22:17,323 --> 00:22:20,960
388
00:22:20,960 --> 00:22:22,600
form a group.
389
00:22:22,600 --> 00:22:25,370
We just defined coset
addition.
390
00:22:25,370 --> 00:22:30,310
And it's easy to check that they
themselves form a group
391
00:22:30,310 --> 00:22:32,980
called the quotient group.
392
00:22:32,980 --> 00:22:37,360
393
00:22:37,360 --> 00:22:45,450
Usually written G slash H and
pronounced G mod H. And we can
394
00:22:45,450 --> 00:22:47,570
can mod out anything.
395
00:22:47,570 --> 00:22:51,190
We do the arithmetic in these
quotient groups by modding out
396
00:22:51,190 --> 00:22:54,560
any elements of H.
397
00:22:54,560 --> 00:22:56,030
And let's take an example.
398
00:22:56,030 --> 00:22:57,280
Here's a good one.
399
00:22:57,280 --> 00:23:04,540
400
00:23:04,540 --> 00:23:13,980
For example, the cosets
of 2Z in Z, namely,
401
00:23:13,980 --> 00:23:18,425
2Z and 2Z plus 1.
402
00:23:18,425 --> 00:23:24,050
403
00:23:24,050 --> 00:23:25,730
Under coset addition.
404
00:23:25,730 --> 00:23:28,142
What is coset addition here?
405
00:23:28,142 --> 00:23:36,330
If I add any even integer to
any even integer, I get an
406
00:23:36,330 --> 00:23:38,510
even integer.
407
00:23:38,510 --> 00:23:41,455
Any odd to even, I get
an odd integer.
408
00:23:41,455 --> 00:23:44,880
Odd to odd gives that.
409
00:23:44,880 --> 00:23:50,370
I mean, odd to even gives that,
and odd to odd gives me
410
00:23:50,370 --> 00:23:52,550
back evens again.
411
00:23:52,550 --> 00:23:55,660
So that's the addition table.
412
00:23:55,660 --> 00:23:59,690
The subgroup itself,
x is the identity.
413
00:23:59,690 --> 00:24:08,660
This is clearly isomorphic to
Z2 with this addition table.
414
00:24:08,660 --> 00:24:12,420
415
00:24:12,420 --> 00:24:16,920
In fact, this is a very good way
of constructing the cyclic
416
00:24:16,920 --> 00:24:20,660
group Z2, or more
generally, Zn.
417
00:24:20,660 --> 00:24:27,920
So this would be called
Z mod 2Z.
418
00:24:27,920 --> 00:24:34,850
And it's isomorphic to Z2, or
in general, Z mod nZ is
419
00:24:34,850 --> 00:24:36,100
isomorphic to Zn.
420
00:24:36,100 --> 00:24:38,800
421
00:24:38,800 --> 00:24:44,150
A very good way of thinking of
Zn is as residue classes or
422
00:24:44,150 --> 00:24:45,780
equivalence classes, modulo n.
423
00:24:45,780 --> 00:24:51,810
424
00:24:51,810 --> 00:24:54,180
The cosets of nZ.
425
00:24:54,180 --> 00:25:00,880
are nZ itself, nZ plus 1,
nZ plus 2, up to nZ
426
00:25:00,880 --> 00:25:05,030
plus n minus 1.
427
00:25:05,030 --> 00:25:08,780
And if you add them together,
they follow the rules of mod n
428
00:25:08,780 --> 00:25:09,540
arithmetic.
429
00:25:09,540 --> 00:25:11,970
If you just add the residues
together and
430
00:25:11,970 --> 00:25:13,500
then reduce mod n.
431
00:25:13,500 --> 00:25:19,040
So we can think of
Zn as being --
432
00:25:19,040 --> 00:25:21,770
a coset is an equivalence
class.
433
00:25:21,770 --> 00:25:24,160
It's all the elements of the
group that are equivalent,
434
00:25:24,160 --> 00:25:29,830
modular of the subgroup H. Or in
the case of integers, it's
435
00:25:29,830 --> 00:25:32,460
all integers that are
equivalent modulo
436
00:25:32,460 --> 00:25:34,870
the subgroup nZ.
437
00:25:34,870 --> 00:25:39,280
They have the same remainder
after division by n.
438
00:25:39,280 --> 00:25:40,370
They have the same residue.
439
00:25:40,370 --> 00:25:42,765
These are all equivalence
class notions.
440
00:25:42,765 --> 00:25:45,510
And how do you add them?
441
00:25:45,510 --> 00:25:50,020
You add them in the ordinary
way, and then you take
442
00:25:50,020 --> 00:25:52,120
everything modulo m.
443
00:25:52,120 --> 00:25:56,440
In other words, you do
mod-m arithmetic.
444
00:25:56,440 --> 00:25:57,090
OK.
445
00:25:57,090 --> 00:26:05,030
So I didn't quite go to quotient
groups in the notes,
446
00:26:05,030 --> 00:26:08,130
but perhaps I should have.
447
00:26:08,130 --> 00:26:13,730
Probably I should have, because
this is maybe the most
448
00:26:13,730 --> 00:26:19,500
powerful idea in group theory,
and certainly closely related
449
00:26:19,500 --> 00:26:23,150
to this little bit of number
theory that we're doing in the
450
00:26:23,150 --> 00:26:25,050
integers mod n.
451
00:26:25,050 --> 00:26:28,600
And of course, it has
vastly greater
452
00:26:28,600 --> 00:26:32,390
applications than just that.
453
00:26:32,390 --> 00:26:32,720
OK.
454
00:26:32,720 --> 00:26:37,410
And you could do the same
thing over here.
455
00:26:37,410 --> 00:26:42,910
This is basically doing the
same kind of thing.
456
00:26:42,910 --> 00:26:45,890
457
00:26:45,890 --> 00:26:48,175
But I won't take time
to do that.
458
00:26:48,175 --> 00:26:50,710
459
00:26:50,710 --> 00:26:55,470
So here's another view of the
integers mod-n that may be
460
00:26:55,470 --> 00:26:56,830
helpful as we go forward.
461
00:26:56,830 --> 00:27:00,590
462
00:27:00,590 --> 00:27:01,120
All right.
463
00:27:01,120 --> 00:27:05,740
I think that's all I want
to say about that.
464
00:27:05,740 --> 00:27:07,070
Yeah.
465
00:27:07,070 --> 00:27:07,490
Good.
466
00:27:07,490 --> 00:27:08,040
All right.
467
00:27:08,040 --> 00:27:10,025
Here would be a good place
to start on fields.
468
00:27:10,025 --> 00:27:18,580
469
00:27:18,580 --> 00:27:19,980
OK.
470
00:27:19,980 --> 00:27:21,230
Fields.
471
00:27:21,230 --> 00:27:22,850
472
00:27:22,850 --> 00:27:25,100
Obviously very important
in algebra.
473
00:27:25,100 --> 00:27:28,160
474
00:27:28,160 --> 00:27:30,490
Fields are like groups,
only more so.
475
00:27:30,490 --> 00:27:34,580
Groups are a set of elements
with a single operation, which
476
00:27:34,580 --> 00:27:35,830
we've been calling addition.
477
00:27:35,830 --> 00:27:38,310
478
00:27:38,310 --> 00:27:42,500
A field is a set of elements
with two operations, which
479
00:27:42,500 --> 00:27:45,990
we'll call addition and
multiplication.
480
00:27:45,990 --> 00:27:46,990
So what do we have?
481
00:27:46,990 --> 00:27:50,720
We have a set of elements F.
We're going to be particularly
482
00:27:50,720 --> 00:27:53,030
interested where the
set is finite.
483
00:27:53,030 --> 00:27:55,272
Those are called
finite fields.
484
00:27:55,272 --> 00:28:00,190
And we're going to have two
operations, which I'll
485
00:28:00,190 --> 00:28:04,620
continue to write addition by
simple plus and multiplication
486
00:28:04,620 --> 00:28:07,740
with an asterisk, just to
be very explicit about
487
00:28:07,740 --> 00:28:08,695
everything.
488
00:28:08,695 --> 00:28:11,710
And after a while, you can
write these things as you
489
00:28:11,710 --> 00:28:15,170
would in ordinary arithmetic,
with just ordinary plus and
490
00:28:15,170 --> 00:28:18,550
juxtaposition for
multiplication.
491
00:28:18,550 --> 00:28:21,101
And what are the axioms
of a field?
492
00:28:21,101 --> 00:28:26,870
They're presented in an elegant
way in the notes,
493
00:28:26,870 --> 00:28:31,650
which obviously go back a long
way, but I got from Bob
494
00:28:31,650 --> 00:28:34,780
Gallager, and I like.
495
00:28:34,780 --> 00:28:36,120
All right.
496
00:28:36,120 --> 00:28:39,590
Under addition --
497
00:28:39,590 --> 00:28:42,710
so let's write it this way.
498
00:28:42,710 --> 00:28:47,500
Now, just considering the
addition operation is an
499
00:28:47,500 --> 00:28:50,720
abelian group.
500
00:28:50,720 --> 00:28:51,970
Commutative group.
501
00:28:51,970 --> 00:28:57,370
502
00:28:57,370 --> 00:29:01,470
Which means it has an identity,
and we will continue
503
00:29:01,470 --> 00:29:05,620
to call that identity 0.
504
00:29:05,620 --> 00:29:09,970
Just as we do in the real
field, let's say.
505
00:29:09,970 --> 00:29:10,085
OK.
506
00:29:10,085 --> 00:29:12,540
Think of the real field, if you
like, as a model for all
507
00:29:12,540 --> 00:29:15,050
fields here.
508
00:29:15,050 --> 00:29:16,520
All right.
509
00:29:16,520 --> 00:29:19,130
So that's axiom one.
510
00:29:19,130 --> 00:29:22,250
Axiom two.
511
00:29:22,250 --> 00:29:26,320
If we take the non-zero elements
of the field, which I
512
00:29:26,320 --> 00:29:32,740
write by F star, explicitly
that's F not including 0 --
513
00:29:32,740 --> 00:29:35,295
514
00:29:35,295 --> 00:29:38,550
not a very good notation,
but I'll use it --
515
00:29:38,550 --> 00:29:45,640
and the multiplication
operation, that, too is an
516
00:29:45,640 --> 00:29:46,890
abelian group.
517
00:29:46,890 --> 00:29:52,590
518
00:29:52,590 --> 00:29:56,390
So this, of course, is why we
spent a little time on groups,
519
00:29:56,390 --> 00:30:00,850
abelian groups, so we'd
eventually be able to deal
520
00:30:00,850 --> 00:30:02,300
with fields.
521
00:30:02,300 --> 00:30:06,000
And its identity is called 1.
522
00:30:06,000 --> 00:30:10,200
523
00:30:10,200 --> 00:30:12,300
Meaning that under
multiplication, 1 times
524
00:30:12,300 --> 00:30:14,445
anything is equal to itself.
525
00:30:14,445 --> 00:30:18,030
526
00:30:18,030 --> 00:30:24,050
And then we have something
about how the operations
527
00:30:24,050 --> 00:30:35,450
distribute, the usual
distributive law that A times
528
00:30:35,450 --> 00:30:44,890
B plus C is equal to A times B
plus D times C, where I've
529
00:30:44,890 --> 00:30:46,670
written out all of these.
530
00:30:46,670 --> 00:30:51,290
So this is how addition and
multiplication interact again
531
00:30:51,290 --> 00:30:53,655
in a way that you're accustomed
to, and after a
532
00:30:53,655 --> 00:30:55,240
while, you don't need to write
all these parentheses.
533
00:30:55,240 --> 00:30:58,770
534
00:30:58,770 --> 00:30:58,930
OK.
535
00:30:58,930 --> 00:31:03,090
So that's actually almost a
simpler set of axioms than for
536
00:31:03,090 --> 00:31:05,020
groups, once we understand
the group axioms.
537
00:31:05,020 --> 00:31:07,830
538
00:31:07,830 --> 00:31:10,310
And so let's check.
539
00:31:10,310 --> 00:31:13,920
Is the real field, is the set
of all real numbers under
540
00:31:13,920 --> 00:31:15,500
ordinary real addition and
541
00:31:15,500 --> 00:31:19,955
multiplication, is that a field?
542
00:31:19,955 --> 00:31:24,840
543
00:31:24,840 --> 00:31:26,040
What do we have to check?
544
00:31:26,040 --> 00:31:30,730
We have to check that under
addition, we're going to take
545
00:31:30,730 --> 00:31:33,240
the additive identity
as being equal to 0.
546
00:31:33,240 --> 00:31:36,420
547
00:31:36,420 --> 00:31:41,800
Under our reduced set of group
axioms, the main thing we have
548
00:31:41,800 --> 00:31:46,060
to check is if we add any real
number to the reals, we get
549
00:31:46,060 --> 00:31:49,075
the reals again and the
one-to-one correspondence is
550
00:31:49,075 --> 00:31:50,355
the permutation.
551
00:31:50,355 --> 00:31:52,940
Is that correct?
552
00:31:52,940 --> 00:31:55,280
Yes, it is.
553
00:31:55,280 --> 00:31:59,840
And so this is OK.
554
00:31:59,840 --> 00:32:03,540
Now under multiplication, if
we take the non-zero real
555
00:32:03,540 --> 00:32:07,010
numbers, here's the question.
556
00:32:07,010 --> 00:32:09,580
557
00:32:09,580 --> 00:32:17,240
If I have some alpha not equal
to 0, is alpha times the
558
00:32:17,240 --> 00:32:21,570
reals, not including 0 -- the
non-zero real numbers --
559
00:32:21,570 --> 00:32:23,250
equal to R star?
560
00:32:23,250 --> 00:32:27,180
And here I'm really implying a
one-to-one correspondence.
561
00:32:27,180 --> 00:32:29,670
So I might write it
more that way.
562
00:32:29,670 --> 00:32:33,730
563
00:32:33,730 --> 00:32:39,150
I pose this question rather
abstractly, but you can easily
564
00:32:39,150 --> 00:32:40,790
convince yourself
that it's true.
565
00:32:40,790 --> 00:32:44,270
566
00:32:44,270 --> 00:32:48,600
Any non-zero number, if I
multiply it by any non-zero
567
00:32:48,600 --> 00:32:51,270
number, I get a non-zero
number.
568
00:32:51,270 --> 00:32:53,480
Is the correspondence
one to one?
569
00:32:53,480 --> 00:32:59,820
Yes, because I can divide out
this number and get alpha.
570
00:32:59,820 --> 00:33:02,810
571
00:33:02,810 --> 00:33:13,150
So alpha x on R star by
multiplication to give R star
572
00:33:13,150 --> 00:33:16,570
again, and this is a one-to-one
correspondence.
573
00:33:16,570 --> 00:33:19,350
But it's obvious why I have
to leave out 0, right?
574
00:33:19,350 --> 00:33:21,880
0 times any real number is 0.
575
00:33:21,880 --> 00:33:29,140
So at O, R star is simply
equal to zero set.
576
00:33:29,140 --> 00:33:32,360
So we always have to leave out
0 from multiplication.
577
00:33:32,360 --> 00:33:34,350
0 doesn't have an inverse.
578
00:33:34,350 --> 00:33:36,740
Everything else does
have an inverse.
579
00:33:36,740 --> 00:33:38,750
Under the standard group
operations, that's what we
580
00:33:38,750 --> 00:33:40,890
have to check.
581
00:33:40,890 --> 00:33:45,850
That would be the alternate
question, does every non-zero
582
00:33:45,850 --> 00:33:47,780
real number have an inverse?
583
00:33:47,780 --> 00:33:49,675
That's easier to see,
the answer is yes.
584
00:33:49,675 --> 00:33:51,810
Multiplicative inverse.
585
00:33:51,810 --> 00:33:55,568
Inverse of alpha is
1 over alpha.
586
00:33:55,568 --> 00:33:57,903
AUDIENCE: [INAUDIBLE]
587
00:33:57,903 --> 00:33:58,530
PROFESSOR: Yes.
588
00:33:58,530 --> 00:34:02,050
I've used the alternative set
of axioms, including the
589
00:34:02,050 --> 00:34:03,300
permutation property.
590
00:34:03,300 --> 00:34:08,940
591
00:34:08,940 --> 00:34:11,020
To check whether there's an
abelian group, I've asked if
592
00:34:11,020 --> 00:34:13,929
alpha R star is the permutation
of R star.
593
00:34:13,929 --> 00:34:18,770
And without going through
details, I claim it is.
594
00:34:18,770 --> 00:34:20,020
Thank you.
595
00:34:20,020 --> 00:34:22,620
596
00:34:22,620 --> 00:34:23,170
All right.
597
00:34:23,170 --> 00:34:25,159
So we checked that.
598
00:34:25,159 --> 00:34:27,750
Of course, the distributive
law holds.
599
00:34:27,750 --> 00:34:33,670
So the real field is a field,
which you probably were
600
00:34:33,670 --> 00:34:35,115
willing to accept on
faith, anyway.
601
00:34:35,115 --> 00:34:37,909
602
00:34:37,909 --> 00:34:40,500
Similarly, you go through
exactly the same arguments for
603
00:34:40,500 --> 00:34:41,750
the complex field.
604
00:34:41,750 --> 00:34:45,330
605
00:34:45,330 --> 00:34:48,060
What about the binary field?
606
00:34:48,060 --> 00:34:51,449
We think we understand
that by now.
607
00:34:51,449 --> 00:34:59,740
Here the operations are mod-2
addition and mod-2
608
00:34:59,740 --> 00:35:00,905
multiplication.
609
00:35:00,905 --> 00:35:04,600
I've written down explicitly
the addition and
610
00:35:04,600 --> 00:35:05,850
multiplication tables.
611
00:35:05,850 --> 00:35:11,410
612
00:35:11,410 --> 00:35:16,701
Under addition, we simply
have Z2 again.
613
00:35:16,701 --> 00:35:19,347
614
00:35:19,347 --> 00:35:20,220
F2.
615
00:35:20,220 --> 00:35:23,446
The additive group of
F2 is simply Z2.
616
00:35:23,446 --> 00:35:27,030
We forget about multiplication.
617
00:35:27,030 --> 00:35:31,220
We've seen quite a few times
now that that's a group.
618
00:35:31,220 --> 00:35:35,580
Under multiplication, what are
the non-zero elements of F2?
619
00:35:35,580 --> 00:35:39,910
620
00:35:39,910 --> 00:35:41,160
Just one element. one.
621
00:35:41,160 --> 00:35:43,870
622
00:35:43,870 --> 00:35:46,230
This includes the identity?
623
00:35:46,230 --> 00:35:48,250
Yes.
624
00:35:48,250 --> 00:35:50,245
Is it a group?
625
00:35:50,245 --> 00:35:50,660
Yeah.
626
00:35:50,660 --> 00:35:51,750
It's a trivial group.
627
00:35:51,750 --> 00:35:53,160
1 times 1 equals 1.
628
00:35:53,160 --> 00:35:56,620
629
00:35:56,620 --> 00:36:00,060
1 under multiplication is
isomorphic to the trivial
630
00:36:00,060 --> 00:36:02,630
group 0 under addition.
631
00:36:02,630 --> 00:36:07,720
Its group table is
1 times 1 is 1.
632
00:36:07,720 --> 00:36:11,750
633
00:36:11,750 --> 00:36:13,110
Sure enough.
634
00:36:13,110 --> 00:36:15,560
That's the identity
permutation.
635
00:36:15,560 --> 00:36:17,482
Sometimes when things get
too trivial, it's a
636
00:36:17,482 --> 00:36:18,170
little hard to check.
637
00:36:18,170 --> 00:36:20,760
But yes.
638
00:36:20,760 --> 00:36:25,180
And distributive is
easy to check.
639
00:36:25,180 --> 00:36:26,430
OK?
640
00:36:26,430 --> 00:36:28,980
641
00:36:28,980 --> 00:36:36,840
So that's all it takes
to define a field.
642
00:36:36,840 --> 00:36:40,600
Of course, by the inverse
property, when we have
643
00:36:40,600 --> 00:36:44,940
addition, this also implies an
inverse and a subtraction
644
00:36:44,940 --> 00:36:49,260
operation and a cancellation
and additive identities.
645
00:36:49,260 --> 00:36:55,350
We have a field element on both
sides of a plus b equals
646
00:36:55,350 --> 00:36:58,620
a plus c, then b plus
b equals c.
647
00:36:58,620 --> 00:37:00,890
That's what I mean
by cancellation.
648
00:37:00,890 --> 00:37:04,010
Similarly under multiplication,
we get a
649
00:37:04,010 --> 00:37:05,950
multiplicative inverse.
650
00:37:05,950 --> 00:37:11,320
1 over alpha, for any alpha in
F. We, therefore, are able to
651
00:37:11,320 --> 00:37:14,490
define division.
652
00:37:14,490 --> 00:37:21,880
And we have cancellation for
multiplicative identity.
653
00:37:21,880 --> 00:37:25,580
So we immediately get a lot from
these group properties.
654
00:37:25,580 --> 00:37:27,765
We get all the properties
you expect of fields.
655
00:37:27,765 --> 00:37:30,700
You can add, subtract, multiply,
or divide, all in
656
00:37:30,700 --> 00:37:32,670
the usual way that we do
over the real field.
657
00:37:32,670 --> 00:37:36,900
658
00:37:36,900 --> 00:37:38,150
OK.
659
00:37:38,150 --> 00:37:40,192
660
00:37:40,192 --> 00:37:41,530
Let's stay over here.
661
00:37:41,530 --> 00:37:48,480
662
00:37:48,480 --> 00:37:51,736
I think my next topic
is prime fields.
663
00:37:51,736 --> 00:37:59,990
664
00:37:59,990 --> 00:38:00,190
Yes.
665
00:38:00,190 --> 00:38:01,880
So prime fields.
666
00:38:01,880 --> 00:38:04,490
When we talked about the
factorization properties of
667
00:38:04,490 --> 00:38:08,630
the integers, we talked
about primes p.
668
00:38:08,630 --> 00:38:17,610
And now I'm going to talk about
Fp is going to be a
669
00:38:17,610 --> 00:38:20,990
field with a finite number
of elements where
670
00:38:20,990 --> 00:38:24,190
the number is a prime.
671
00:38:24,190 --> 00:38:27,140
So what are the elements in
this field going to be?
672
00:38:27,140 --> 00:38:34,890
Are they simply going to be 0,
1 up through p minus 1 again,
673
00:38:34,890 --> 00:38:40,240
the same elements as were in
the cyclic group with p
674
00:38:40,240 --> 00:38:44,180
elements where I'm restricting
m now to be a prime p?
675
00:38:44,180 --> 00:38:47,100
676
00:38:47,100 --> 00:38:54,120
And for my addition operation
and my multiplication
677
00:38:54,120 --> 00:38:58,650
operation, I'm going to just
let these be mod-p addition
678
00:38:58,650 --> 00:39:00,613
and multiplication now.
679
00:39:00,613 --> 00:39:05,930
680
00:39:05,930 --> 00:39:08,680
And I claim that this
is a field.
681
00:39:08,680 --> 00:39:14,810
682
00:39:14,810 --> 00:39:19,440
So actually, the proof follows
very close to what I
683
00:39:19,440 --> 00:39:21,620
just did for F2.
684
00:39:21,620 --> 00:39:22,905
F2 is a model for this.
685
00:39:22,905 --> 00:39:25,840
686
00:39:25,840 --> 00:39:28,720
But it's a little harder
to check this.
687
00:39:28,720 --> 00:39:35,710
Under a, under the addition
operation, Fp really is just
688
00:39:35,710 --> 00:39:39,360
Zp again, so that's OK.
689
00:39:39,360 --> 00:39:41,840
That's an abelian group.
690
00:39:41,840 --> 00:39:43,090
Zmod-p.
691
00:39:43,090 --> 00:39:45,010
692
00:39:45,010 --> 00:39:47,120
Or the quotient group,
Zmod-pZ.
693
00:39:47,120 --> 00:39:49,790
694
00:39:49,790 --> 00:39:58,110
We can also again think of this
as Zmod-pZ, if we want.
695
00:39:58,110 --> 00:40:01,720
696
00:40:01,720 --> 00:40:05,080
So everything is going to become
mod-Z. That's a very
697
00:40:05,080 --> 00:40:06,720
useful way of thinking of it.
698
00:40:06,720 --> 00:40:09,810
So we're really thinking of
these as remainders or
699
00:40:09,810 --> 00:40:18,240
representatives of the residue
classes of pZ in Z. This is
700
00:40:18,240 --> 00:40:24,610
pZ, this is pZ plus 1 up
to Z minus 1, up to pZ
701
00:40:24,610 --> 00:40:26,450
plus p minus 1.
702
00:40:26,450 --> 00:40:29,790
This is the same
as pZ minus 1.
703
00:40:29,790 --> 00:40:30,190
OK.
704
00:40:30,190 --> 00:40:34,920
The real question is, if we take
the non-zero elements of
705
00:40:34,920 --> 00:40:42,915
Fp, is this closed?
706
00:40:42,915 --> 00:40:46,110
707
00:40:46,110 --> 00:40:50,860
And does every element
have an inverse?
708
00:40:50,860 --> 00:40:54,130
Or equivalently, when we
multiply by a particular
709
00:40:54,130 --> 00:40:56,400
element, do we just get
a permutation of this?
710
00:40:56,400 --> 00:41:00,250
711
00:41:00,250 --> 00:41:02,280
The reason that p has
to be a prime --
712
00:41:02,280 --> 00:41:08,040
let's suppose we take two of
these things, a and b, and we
713
00:41:08,040 --> 00:41:09,290
multiply them.
714
00:41:09,290 --> 00:41:11,820
715
00:41:11,820 --> 00:41:13,305
What's the multiplicative
rule?
716
00:41:13,305 --> 00:41:21,800
717
00:41:21,800 --> 00:41:26,095
a times b is just ab mod-p.
718
00:41:26,095 --> 00:41:29,380
That's what I defined
multiplication as.
719
00:41:29,380 --> 00:41:33,660
Now the question is, could
that possibly be 0?
720
00:41:33,660 --> 00:41:38,530
Which is the same as saying,
could a times b be a multiple
721
00:41:38,530 --> 00:41:43,760
of p, where a and b are taken
from the non-zero
722
00:41:43,760 --> 00:41:46,460
elements of the field?
723
00:41:46,460 --> 00:41:49,970
And here, because p is a prime,
it's clear that you
724
00:41:49,970 --> 00:41:55,410
can't multiply two non-zero
numbers which are less than p
725
00:41:55,410 --> 00:41:57,870
and get a multiple
of the prime p.
726
00:41:57,870 --> 00:42:02,120
727
00:42:02,120 --> 00:42:08,190
If p were not a prime,
then you could.
728
00:42:08,190 --> 00:42:12,570
If we took n equals 10 again,
let's say, and we multiplied 2
729
00:42:12,570 --> 00:42:18,390
times 5 from the 10 elements
of these residue classes, 2
730
00:42:18,390 --> 00:42:25,370
times 5 is, in fact, equal to
0 mod-10, and therefore Fp
731
00:42:25,370 --> 00:42:30,040
star, or F10 star, would
not be closed under
732
00:42:30,040 --> 00:42:30,930
multiplication.
733
00:42:30,930 --> 00:42:32,140
We would get a 0.
734
00:42:32,140 --> 00:42:35,550
But in this case, we easily
prove, because it's a prime,
735
00:42:35,550 --> 00:42:37,930
that it's not equal to 0.
736
00:42:37,930 --> 00:42:42,930
And therefore it's in Fp star,
so it is closed under
737
00:42:42,930 --> 00:42:44,180
multiplication.
738
00:42:44,180 --> 00:42:46,560
739
00:42:46,560 --> 00:42:50,050
And the other thing we have to
check is that it's one-to-one.
740
00:42:50,050 --> 00:43:02,080
In other words, can a star b
equal to a star c, and by the
741
00:43:02,080 --> 00:43:05,830
cancellation property,
which holds in --
742
00:43:05,830 --> 00:43:09,620
743
00:43:09,620 --> 00:43:09,930
Sorry.
744
00:43:09,930 --> 00:43:13,450
We've got to establish the
cancellation property holds
745
00:43:13,450 --> 00:43:18,140
under mod-b arithmetic, but
it does, and so we get the
746
00:43:18,140 --> 00:43:20,900
cancellation property, that
this is true if and
747
00:43:20,900 --> 00:43:23,340
only if b equals c.
748
00:43:23,340 --> 00:43:28,530
So in other words, as we run
through all of these multiples
749
00:43:28,530 --> 00:43:31,910
for any particular alpha,
we're going to get a
750
00:43:31,910 --> 00:43:32,690
permutation.
751
00:43:32,690 --> 00:43:34,290
We need to get the
same set back.
752
00:43:34,290 --> 00:43:35,820
Everything is finite.
753
00:43:35,820 --> 00:43:38,960
We're going to get a bunch of
distinct elements of the same
754
00:43:38,960 --> 00:43:41,310
size as the set itself.
755
00:43:41,310 --> 00:43:45,920
Therefore, it has to
be the set again.
756
00:43:45,920 --> 00:43:47,910
I haven't said that
very well again.
757
00:43:47,910 --> 00:43:48,980
That's why we have notes.
758
00:43:48,980 --> 00:43:53,090
It's written up correctly
in the notes.
759
00:43:53,090 --> 00:43:56,390
But we have basically checked
everything that we need to
760
00:43:56,390 --> 00:44:01,580
check, showed that Fp star
is an abelian group under
761
00:44:01,580 --> 00:44:06,690
multiplication when p is a
prime, and clearly not when p
762
00:44:06,690 --> 00:44:08,085
is not a prime.
763
00:44:08,085 --> 00:44:10,065
AUDIENCE: [INAUDIBLE]
764
00:44:10,065 --> 00:44:13,400
the inverse, there
is inverse of a?
765
00:44:13,400 --> 00:44:14,650
PROFESSOR: Yes.
766
00:44:14,650 --> 00:44:16,610
767
00:44:16,610 --> 00:44:19,950
But basically, we have to prove
that if I take any of
768
00:44:19,950 --> 00:44:25,790
these, if I take a particular
one, say, alpha, and multiply
769
00:44:25,790 --> 00:44:31,990
times all of them in Fp star,
that I'm just going to get Fp
770
00:44:31,990 --> 00:44:34,080
star again.
771
00:44:34,080 --> 00:44:38,940
And to prove that, I have to
prove that alpha times a is
772
00:44:38,940 --> 00:44:47,710
not equal to alpha times
b if a not equal to b.
773
00:44:47,710 --> 00:44:49,700
That's all I need
to prove, right?
774
00:44:49,700 --> 00:44:52,370
And that comes from the
properties of mod-p
775
00:44:52,370 --> 00:44:53,000
arithmetic.
776
00:44:53,000 --> 00:44:54,865
That is what is to be proved.
777
00:44:54,865 --> 00:44:58,232
I need to use mod-p arithmetic
to prove that.
778
00:44:58,232 --> 00:44:59,482
AUDIENCE: [INAUDIBLE]
779
00:44:59,482 --> 00:45:03,080
780
00:45:03,080 --> 00:45:03,650
PROFESSOR: Oh.
781
00:45:03,650 --> 00:45:08,050
I have to check the identity
is in here.
782
00:45:08,050 --> 00:45:09,630
The identity is in here.
783
00:45:09,630 --> 00:45:10,230
It has one.
784
00:45:10,230 --> 00:45:10,670
I'm sorry.
785
00:45:10,670 --> 00:45:12,175
I should have checked
that, too.
786
00:45:12,175 --> 00:45:15,060
But 1 is the identity
for multiplication.
787
00:45:15,060 --> 00:45:19,380
And then from this property,
since we multiply alpha p
788
00:45:19,380 --> 00:45:22,390
star, we get Fp star again.
789
00:45:22,390 --> 00:45:25,080
That includes one.
790
00:45:25,080 --> 00:45:25,490
All right?
791
00:45:25,490 --> 00:45:29,940
So it's got to be one of these
guys which, times alpha, gives
792
00:45:29,940 --> 00:45:33,200
1, and that shows the existence
of an inverse.
793
00:45:33,200 --> 00:45:38,080
So you can do it any
way you want.
794
00:45:38,080 --> 00:45:42,130
But the key to the proof is to
prove this, and that's why I
795
00:45:42,130 --> 00:45:45,720
focused on the permutation
property.
796
00:45:45,720 --> 00:45:47,850
Permutation property is
really what you prove
797
00:45:47,850 --> 00:45:49,100
to demonstrate this.
798
00:45:49,100 --> 00:45:53,390
799
00:45:53,390 --> 00:45:54,150
OK?
800
00:45:54,150 --> 00:45:55,900
Good.
801
00:45:55,900 --> 00:45:58,670
Everyone seems to be following
closely here.
802
00:45:58,670 --> 00:46:00,335
Any further questions?
803
00:46:00,335 --> 00:46:03,020
This is important, because
we've got our
804
00:46:03,020 --> 00:46:05,360
first finite field.
805
00:46:05,360 --> 00:46:09,640
The integers mod-p are a finite
field of size p for any
806
00:46:09,640 --> 00:46:11,110
prime state.
807
00:46:11,110 --> 00:46:15,540
We've got F2, F3, F5,
F7, and so forth.
808
00:46:15,540 --> 00:46:19,830
809
00:46:19,830 --> 00:46:22,370
OK.
810
00:46:22,370 --> 00:46:23,855
Further on this subject.
811
00:46:23,855 --> 00:46:31,620
812
00:46:31,620 --> 00:46:34,670
We have two closely related
propositions.
813
00:46:34,670 --> 00:46:57,100
One, every finite field with
prime p elements is
814
00:46:57,100 --> 00:46:58,840
isomorphic to Fp.
815
00:46:58,840 --> 00:47:02,230
816
00:47:02,230 --> 00:47:05,690
So if you give me a finite
field, you tell me it has p
817
00:47:05,690 --> 00:47:09,980
elements, I'll show you that
it basically has the same
818
00:47:09,980 --> 00:47:11,940
addition and multiplication
tables with relabeling.
819
00:47:11,940 --> 00:47:16,460
820
00:47:16,460 --> 00:47:29,060
And secondly, every finite field
with an arbitrary number
821
00:47:29,060 --> 00:47:38,970
of elements, for every finite
field, the integers of the
822
00:47:38,970 --> 00:47:52,330
field form a prime field for
some P. You understand my
823
00:47:52,330 --> 00:47:55,250
abbreviations.
824
00:47:55,250 --> 00:47:58,010
And the proofs of these are
very closely related.
825
00:47:58,010 --> 00:48:05,290
What do I mean by the integers
of a field, of a finite field?
826
00:48:05,290 --> 00:48:13,530
827
00:48:13,530 --> 00:48:14,150
OK.
828
00:48:14,150 --> 00:48:20,270
Well, let's start from the
very most basic thing.
829
00:48:20,270 --> 00:48:21,590
What do we know?
830
00:48:21,590 --> 00:48:24,730
We know that the field contains
0 and 1, and those
831
00:48:24,730 --> 00:48:27,760
are going to be two of the
integers of the field.
832
00:48:27,760 --> 00:48:32,120
So 0 and 1 are in F.
833
00:48:32,120 --> 00:48:39,060
Let's use the closure
under addition.
834
00:48:39,060 --> 00:48:44,995
Clearly 1 plus 1 is in F. We're
going to call that 2.
835
00:48:44,995 --> 00:48:47,630
836
00:48:47,630 --> 00:48:54,420
1 plus 1 plus 1 is in F. We're
going to call that 3.
837
00:48:54,420 --> 00:48:55,670
And so forth.
838
00:48:55,670 --> 00:48:59,620
839
00:48:59,620 --> 00:49:04,480
And of course, since the field
is finite, eventually this is
840
00:49:04,480 --> 00:49:07,240
going to have to repeat.
841
00:49:07,240 --> 00:49:16,700
And from the fact it repeats,
you're basically going to show
842
00:49:16,700 --> 00:49:19,920
that at some point, one
of these is going
843
00:49:19,920 --> 00:49:22,000
to be equal to 0.
844
00:49:22,000 --> 00:49:25,560
So there's going to be some n.
845
00:49:25,560 --> 00:49:27,690
The first repeat is
going to be n
846
00:49:27,690 --> 00:49:38,430
equal to 0 in F. OK.
847
00:49:38,430 --> 00:49:39,870
So that's what I mean
by the integers.
848
00:49:39,870 --> 00:49:45,490
849
00:49:45,490 --> 00:49:59,260
The integers clearly form a
subgroup of the additive group
850
00:49:59,260 --> 00:50:02,940
of F, to form a subgroup
under addition.
851
00:50:02,940 --> 00:50:09,540
852
00:50:09,540 --> 00:50:12,806
And in fact, a cyclic
subgroup.
853
00:50:12,806 --> 00:50:18,800
I'm skipping over some of the
details here, but that's a
854
00:50:18,800 --> 00:50:23,240
claim at this point that I
haven't really demonstrated.
855
00:50:23,240 --> 00:50:27,890
But just from a subgroup
property, let's attack number
856
00:50:27,890 --> 00:50:30,040
one up here.
857
00:50:30,040 --> 00:50:35,030
Suppose we have a field with p
elements, and the additive
858
00:50:35,030 --> 00:50:38,310
group of the field
has p elements.
859
00:50:38,310 --> 00:50:42,110
It consists of the
same elements.
860
00:50:42,110 --> 00:50:46,000
And by Lagrange's theorem, what
are the possible orders
861
00:50:46,000 --> 00:50:47,485
of that subgroup?
862
00:50:47,485 --> 00:50:50,090
863
00:50:50,090 --> 00:50:52,780
What are the possible number of
elements in that subgroup,
864
00:50:52,780 --> 00:50:54,980
the sizes of the subgroup?
865
00:50:54,980 --> 00:50:56,315
The order of a group
is its size.
866
00:50:56,315 --> 00:50:59,670
867
00:50:59,670 --> 00:51:01,460
Well, it has to divide p.
868
00:51:01,460 --> 00:51:04,095
There aren't many things
that divide a prime p.
869
00:51:04,095 --> 00:51:06,710
There's 1 and there's p.
870
00:51:06,710 --> 00:51:07,140
OK?
871
00:51:07,140 --> 00:51:12,960
So the subgroup either has
a single element or
872
00:51:12,960 --> 00:51:15,530
it's all of the group.
873
00:51:15,530 --> 00:51:18,040
874
00:51:18,040 --> 00:51:21,020
If there's a single element
-- let's to keep an
875
00:51:21,020 --> 00:51:24,520
open mind here --
876
00:51:24,520 --> 00:51:31,350
then what that means is that
if I take G and I add it to
877
00:51:31,350 --> 00:51:33,750
itself, since it's a subgroup,
it has to give an
878
00:51:33,750 --> 00:51:35,360
element of the group.
879
00:51:35,360 --> 00:51:39,140
But there is only one element
of the group.
880
00:51:39,140 --> 00:51:43,590
Let's say G is the single
element in this subgroup.
881
00:51:43,590 --> 00:51:45,310
I guess it could only be 1.
882
00:51:45,310 --> 00:51:46,750
Let's start out with one.
883
00:51:46,750 --> 00:51:50,480
So suppose one is the only
element of the subgroup.
884
00:51:50,480 --> 00:51:55,560
Then I get the equation 1 plus
1 equals 1, which by
885
00:51:55,560 --> 00:52:01,090
cancellation implies
that 1 equals 0.
886
00:52:01,090 --> 00:52:03,250
OK, well, that can't be true.
887
00:52:03,250 --> 00:52:06,490
In a field the multiplicative
identity is not
888
00:52:06,490 --> 00:52:08,550
the additive identity.
889
00:52:08,550 --> 00:52:13,610
So that can't be true.
890
00:52:13,610 --> 00:52:17,370
That would only be true if we
had a field with one element,
891
00:52:17,370 --> 00:52:21,340
and fields implicitly always
have at least two
892
00:52:21,340 --> 00:52:23,470
elements, 0 and 1.
893
00:52:23,470 --> 00:52:26,300
F2 is the smallest
finite field.
894
00:52:26,300 --> 00:52:29,390
I suppose we could set up a
single element that sort of
895
00:52:29,390 --> 00:52:32,590
satisfies all these axioms,
but then, what is the
896
00:52:32,590 --> 00:52:34,100
multiplicative group?
897
00:52:34,100 --> 00:52:35,460
All right.
898
00:52:35,460 --> 00:52:37,660
So this can't happen.
899
00:52:37,660 --> 00:52:41,710
So that means this subgroup
has to have p elements.
900
00:52:41,710 --> 00:52:44,370
It has to consist of all the
elements of the field.
901
00:52:44,370 --> 00:52:47,310
So that means the integers are
all the elements of the field.
902
00:52:47,310 --> 00:52:50,600
903
00:52:50,600 --> 00:52:58,220
But now the isomorphism, then,
is that this is isomorphic Fp
904
00:52:58,220 --> 00:52:59,470
under the isomorphism.
905
00:52:59,470 --> 00:53:01,560
906
00:53:01,560 --> 00:53:05,500
This corresponds to 2, this
corresponds to 3, and so
907
00:53:05,500 --> 00:53:06,980
forth, in Fp.
908
00:53:06,980 --> 00:53:12,190
You can see, you know, 2 is 1
plus 1, 3 is 1 plus 1 plus 1,
909
00:53:12,190 --> 00:53:16,170
so 2 plus 3 is going
to be 5 1's.
910
00:53:16,170 --> 00:53:18,690
911
00:53:18,690 --> 00:53:22,410
Mod size the field, whenever
this cycles.
912
00:53:22,410 --> 00:53:25,510
So this is going to
have to be p, and
913
00:53:25,510 --> 00:53:27,570
basically, that shows --
914
00:53:27,570 --> 00:53:28,820
AUDIENCE: [INAUDIBLE]
915
00:53:28,820 --> 00:53:30,670
916
00:53:30,670 --> 00:53:33,288
to prove that it is isomorphical
[UNINTELLIGIBLE]
917
00:53:33,288 --> 00:53:37,536
multiplying 1 plus 1 into
1 plus 1 plus 1.
918
00:53:37,536 --> 00:53:38,810
But it is typical --
919
00:53:38,810 --> 00:53:40,240
PROFESSOR: I really
have only used the
920
00:53:40,240 --> 00:53:41,380
additive property here.
921
00:53:41,380 --> 00:53:43,884
I don't think multiplication
enters into it.
922
00:53:43,884 --> 00:53:47,300
923
00:53:47,300 --> 00:53:52,500
OK, here's where the
multiplicative
924
00:53:52,500 --> 00:53:53,610
property adds in.
925
00:53:53,610 --> 00:53:59,320
I have to prove not only that
this is isomorphic to Fp as an
926
00:53:59,320 --> 00:54:11,160
additive group, but the
multiplication tables are
927
00:54:11,160 --> 00:54:14,690
isomorphic under the
same relabeling.
928
00:54:14,690 --> 00:54:19,410
So for that, I have to show
that 2 times 3, when I've
929
00:54:19,410 --> 00:54:22,250
defined 3 and 3 this way, gives
me the same result as
930
00:54:22,250 --> 00:54:26,500
multiplying 2 and
3 in F p mod-p.
931
00:54:26,500 --> 00:54:31,530
But again, I could do this just
because sort of mod-p
932
00:54:31,530 --> 00:54:35,840
commutes with addition
and multiplication.
933
00:54:35,840 --> 00:54:42,960
If I multiply 1 plus 1, two 1's
times three 1's, so I'm
934
00:54:42,960 --> 00:54:47,630
going to get six 1's, and that's
exactly what I get in
935
00:54:47,630 --> 00:54:51,290
Fp, reducing everything mod-p.
936
00:54:51,290 --> 00:54:57,500
So I have to check that also
to prove this isomorphism.
937
00:54:57,500 --> 00:55:00,070
And this is done carefully
in the notes.
938
00:55:00,070 --> 00:55:02,850
The distributive law holds
because the distributive law
939
00:55:02,850 --> 00:55:07,090
holds for sums of n 1's.
940
00:55:07,090 --> 00:55:11,120
941
00:55:11,120 --> 00:55:14,680
1 plus 1 times 1 plus 1 plus 1
plus 1, it's going to be the
942
00:55:14,680 --> 00:55:18,255
same regardless of where you
put it in, how you put the
943
00:55:18,255 --> 00:55:20,150
parentheses.
944
00:55:20,150 --> 00:55:20,570
OK.
945
00:55:20,570 --> 00:55:27,230
So with some sorry hand-waving
here, we've basically given
946
00:55:27,230 --> 00:55:29,750
the idea of how to prove this.
947
00:55:29,750 --> 00:55:33,890
It's basically Lagrange that the
additive subgroup has to
948
00:55:33,890 --> 00:55:42,360
be of size 1 or p, and we prove
quickly that actually p
949
00:55:42,360 --> 00:55:44,360
is the only case that works.
950
00:55:44,360 --> 00:55:48,260
And then we extend all the
arithmetic properties by just
951
00:55:48,260 --> 00:55:51,888
observing they'll hold
for 1 plus 1 plus 1.
952
00:55:51,888 --> 00:55:52,382
Yeah?
953
00:55:52,382 --> 00:55:54,852
AUDIENCE: [INAUDIBLE]
954
00:55:54,852 --> 00:55:58,580
the line 1 plus 1 plus
1 like that?
955
00:55:58,580 --> 00:55:59,830
Might we [UNINTELLIGIBLE PHRASE]
956
00:55:59,830 --> 00:56:03,794
957
00:56:03,794 --> 00:56:05,700
1 is equal to 0.
958
00:56:05,700 --> 00:56:08,270
What is actually
[UNINTELLIGIBLE PHRASE]?
959
00:56:08,270 --> 00:56:11,135
Should we state that 1 has
to be different than 0?
960
00:56:11,135 --> 00:56:11,560
PROFESSOR: Yeah.
961
00:56:11,560 --> 00:56:15,360
I guess I could simply get
around that by stating that
962
00:56:15,360 --> 00:56:19,270
the multiplicative identity has
to be different from the
963
00:56:19,270 --> 00:56:20,260
additive identity.
964
00:56:20,260 --> 00:56:23,710
It clearly follows from this,
and I think I put it as an
965
00:56:23,710 --> 00:56:28,930
exercise, 0 times any group
element has got
966
00:56:28,930 --> 00:56:31,600
to be equal to 0.
967
00:56:31,600 --> 00:56:33,360
So this is how 0 behaves under
968
00:56:33,360 --> 00:56:37,150
multiplication from these axioms.
969
00:56:37,150 --> 00:56:40,520
But 1, as the multiplicative
identity, has to satisfy that
970
00:56:40,520 --> 00:56:45,960
rule, so clearly, 0
cannot equal to 1.
971
00:56:45,960 --> 00:56:49,250
Unless, in some trivial sense,
there is only one element in
972
00:56:49,250 --> 00:56:52,580
the groups if there's any
non-zero element.
973
00:56:52,580 --> 00:56:56,860
So this implies that 0
is not equal to 1.
974
00:56:56,860 --> 00:56:59,092
Just could have included
that as an axiom.
975
00:56:59,092 --> 00:57:02,068
976
00:57:02,068 --> 00:57:02,564
Yeah?
977
00:57:02,564 --> 00:57:07,028
AUDIENCE: Assume a
and b [INAUDIBLE]
978
00:57:07,028 --> 00:57:08,020
PROFESSOR: Excuse me?
979
00:57:08,020 --> 00:57:11,530
AUDIENCE: Assume a and
b, [UNINTELLIGIBLE]
980
00:57:11,530 --> 00:57:13,620
does not include 0?
981
00:57:13,620 --> 00:57:14,870
PROFESSOR: Correct.
982
00:57:14,870 --> 00:57:17,060
983
00:57:17,060 --> 00:57:17,810
Yeah, you're right.
984
00:57:17,810 --> 00:57:18,440
OK.
985
00:57:18,440 --> 00:57:25,155
So it follows from this
that 1 is not 0.
986
00:57:25,155 --> 00:57:26,100
Yeah.
987
00:57:26,100 --> 00:57:28,740
I'm sorry I don't personally
have a lot of patience for
988
00:57:28,740 --> 00:57:30,810
these fine details.
989
00:57:30,810 --> 00:57:33,430
For mathematicians,
it's important to
990
00:57:33,430 --> 00:57:38,340
keep them all in mind.
991
00:57:38,340 --> 00:57:41,830
But my effort is to make these
propositions plausible enough
992
00:57:41,830 --> 00:57:45,800
so that you can believe them,
and you can go back and read a
993
00:57:45,800 --> 00:57:50,410
real proof and see that the
proof must be correct
994
00:57:50,410 --> 00:57:53,360
intuitively, without just
mechanically checking it.
995
00:57:53,360 --> 00:58:00,070
996
00:58:00,070 --> 00:58:02,410
OK.
997
00:58:02,410 --> 00:58:07,470
Let me just again outline
how this works.
998
00:58:07,470 --> 00:58:09,260
It's very similar.
999
00:58:09,260 --> 00:58:12,800
Again, given any finite field,
if we define the integers of
1000
00:58:12,800 --> 00:58:18,870
the field in this way, we show
that eventually they form a
1001
00:58:18,870 --> 00:58:19,930
cyclic group.
1002
00:58:19,930 --> 00:58:22,735
Their cyclic group
is something that
1003
00:58:22,735 --> 00:58:23,820
has a single generator.
1004
00:58:23,820 --> 00:58:25,630
The generator is 1.
1005
00:58:25,630 --> 00:58:30,880
So eventually it has to cycle
for some number n.
1006
00:58:30,880 --> 00:58:35,740
Now could n be a non-prime?
1007
00:58:35,740 --> 00:58:41,110
No, because this is a field, and
if n were non-prime, then
1008
00:58:41,110 --> 00:58:44,040
we would be able to find two
integers that multiplied
1009
00:58:44,040 --> 00:58:47,960
together gave 0.
1010
00:58:47,960 --> 00:58:53,600
And that's forbidden by the
axioms of the field.
1011
00:58:53,600 --> 00:58:59,970
So the only possibility is that
n is a prime, and in that
1012
00:58:59,970 --> 00:59:05,040
case, we have found what's
called a subfield, a subset of
1013
00:59:05,040 --> 00:59:07,960
the elements of the field which
itself is a field under
1014
00:59:07,960 --> 00:59:09,210
the field axioms.
1015
00:59:09,210 --> 00:59:11,340
1016
00:59:11,340 --> 00:59:14,050
And the field has p elements,
and we already know that every
1017
00:59:14,050 --> 00:59:19,150
finite field with p elements
is isomorphic to Fp.
1018
00:59:19,150 --> 00:59:25,220
So it can only be that the set
of integers is a subfield
1019
00:59:25,220 --> 00:59:28,030
which is isomorphic to
Fp for some prime p.
1020
00:59:28,030 --> 00:59:30,670
1021
00:59:30,670 --> 00:59:31,990
OK?
1022
00:59:31,990 --> 00:59:36,320
So within any finite field,
we're always going to find,
1023
00:59:36,320 --> 00:59:41,090
just by writing out the integers
and seeing how they
1024
00:59:41,090 --> 00:59:44,140
behave under the additive
property that there are
1025
00:59:44,140 --> 00:59:45,390
exactly p of them.
1026
00:59:45,390 --> 00:59:51,280
1027
00:59:51,280 --> 00:59:54,490
So every finite field has
a prime called the
1028
00:59:54,490 --> 00:59:55,276
characteristic.
1029
00:59:55,276 --> 00:59:57,370
The prime characteristic
of the field.
1030
00:59:57,370 --> 00:59:59,760
This is defined as the
characteristic.
1031
00:59:59,760 --> 01:00:03,580
The size of the integer subfield
is the characteristic
1032
01:00:03,580 --> 01:00:05,410
of the field.
1033
01:00:05,410 --> 01:00:11,030
And it has an interesting
property, a
1034
01:00:11,030 --> 01:00:12,280
very important property.
1035
01:00:12,280 --> 01:00:17,480
Suppose we take this p and we
multiply it by any field
1036
01:00:17,480 --> 01:00:21,190
element called data
in the field.
1037
01:00:21,190 --> 01:00:25,310
1038
01:00:25,310 --> 01:00:28,420
By the distributive law,
this is just equal to
1039
01:00:28,420 --> 01:00:32,110
1 plus 1 plus --
1040
01:00:32,110 --> 01:00:33,780
so what do I mean by this?
1041
01:00:33,780 --> 01:00:37,410
I mean beta --
1042
01:00:37,410 --> 01:00:40,980
whenever I write an integer
times a field element, I mean
1043
01:00:40,980 --> 01:00:46,220
beta plus beta plus
so forth, p times.
1044
01:00:46,220 --> 01:00:49,880
1045
01:00:49,880 --> 01:00:56,430
But this is equal to 1 plus 1,
so forth, by the distributive
1046
01:00:56,430 --> 01:01:03,670
law, I guess, times
theta, p times.
1047
01:01:03,670 --> 01:01:04,920
And what is this equal to?
1048
01:01:04,920 --> 01:01:08,260
1049
01:01:08,260 --> 01:01:09,290
This is equal to 0.
1050
01:01:09,290 --> 01:01:13,200
So this is equal to 0 times
beta, which fortunately I just
1051
01:01:13,200 --> 01:01:15,585
told you always must equal 0.
1052
01:01:15,585 --> 01:01:21,360
1053
01:01:21,360 --> 01:01:21,880
OK.
1054
01:01:21,880 --> 01:01:28,840
So the conclusion is that if
we add any field element to
1055
01:01:28,840 --> 01:01:35,860
itself p times, we're going to
get 0 for all beta in the
1056
01:01:35,860 --> 01:01:39,870
field where p is the
characteristic of the field.
1057
01:01:39,870 --> 01:01:43,700
1058
01:01:43,700 --> 01:01:49,560
Now in digital communications,
we're almost always dealing
1059
01:01:49,560 --> 01:01:52,690
with a case where the
characteristic of the field is
1060
01:01:52,690 --> 01:01:53,760
going to be 2.
1061
01:01:53,760 --> 01:01:55,940
The prime subfield is just
going to be the two
1062
01:01:55,940 --> 01:01:57,620
elements 0 and 1.
1063
01:01:57,620 --> 01:02:01,290
1 plus 1 is going to
be equal to 0.
1064
01:02:01,290 --> 01:02:04,450
So subtraction will be
the same as addition.
1065
01:02:04,450 --> 01:02:11,420
And in that particular case, we
will have that the sum beta
1066
01:02:11,420 --> 01:02:14,950
plus beta of any 2 field
elements in a field of
1067
01:02:14,950 --> 01:02:18,930
characteristic two is going
to be equal to 0.
1068
01:02:18,930 --> 01:02:21,910
Just as we had for code words
in binary linear codes.
1069
01:02:21,910 --> 01:02:25,470
1070
01:02:25,470 --> 01:02:28,980
Binary linear codes are not
fields, they're vector spaces,
1071
01:02:28,980 --> 01:02:31,160
but it's a similar
property here.
1072
01:02:31,160 --> 01:02:35,900
You add any element of field of
characteristic 2 to itself,
1073
01:02:35,900 --> 01:02:37,300
and you're going to get 0.
1074
01:02:37,300 --> 01:02:41,450
So this shows that addition is
the same as subtraction.
1075
01:02:41,450 --> 01:02:47,290
Beta equals minus beta in a
field of characteristic 0.
1076
01:02:47,290 --> 01:02:50,370
Which is a little bit
more general.
1077
01:02:50,370 --> 01:02:53,930
1078
01:02:53,930 --> 01:02:54,025
OK.
1079
01:02:54,025 --> 01:03:00,170
So we have some fields now, and
we find these fields are
1080
01:03:00,170 --> 01:03:05,750
the only field of prime size,
and that every finite field
1081
01:03:05,750 --> 01:03:10,700
has an important subfield
and a prime subfield.
1082
01:03:10,700 --> 01:03:17,290
And that has important
properties, consequences for
1083
01:03:17,290 --> 01:03:19,870
the field itself.
1084
01:03:19,870 --> 01:03:20,550
All right.
1085
01:03:20,550 --> 01:03:22,290
I think that's everything
I want to
1086
01:03:22,290 --> 01:03:27,170
say about prime fields.
1087
01:03:27,170 --> 01:03:31,560
Now we go on to the next
important algebraic object,
1088
01:03:31,560 --> 01:03:32,810
polynomials.
1089
01:03:32,810 --> 01:03:37,690
1090
01:03:37,690 --> 01:03:45,160
And again, it's hard to know
just how detailed to be,
1091
01:03:45,160 --> 01:03:51,010
because of course you've all
seen polynomials, and you
1092
01:03:51,010 --> 01:03:54,520
intuitively or formally know
something about their
1093
01:03:54,520 --> 01:03:56,830
algebraic properties,
their factorization
1094
01:03:56,830 --> 01:03:58,630
properties, and so forth.
1095
01:03:58,630 --> 01:04:02,200
So I'm going to go pretty
quickly, and this will be in
1096
01:04:02,200 --> 01:04:03,450
the nature of a review.
1097
01:04:03,450 --> 01:04:07,350
1098
01:04:07,350 --> 01:04:08,600
A polynomial --
1099
01:04:08,600 --> 01:04:12,130
1100
01:04:12,130 --> 01:04:14,020
maybe the simplest way --
1101
01:04:14,020 --> 01:04:16,650
how do you define
a polynomial?
1102
01:04:16,650 --> 01:04:18,670
What does it look like?
1103
01:04:18,670 --> 01:04:20,760
It looks like this.
1104
01:04:20,760 --> 01:04:29,900
F0 plus F1 times x plus F2 times
x squared, so forth,
1105
01:04:29,900 --> 01:04:33,120
plus Fm times x to the m.
1106
01:04:33,120 --> 01:04:36,800
That's what it looks like if
it's a non-zero polynomial.
1107
01:04:36,800 --> 01:04:40,010
1108
01:04:40,010 --> 01:04:40,900
Or even if it's just --
1109
01:04:40,900 --> 01:04:44,800
you could consider all 0
coefficients to be the zero
1110
01:04:44,800 --> 01:04:45,420
polynomial.
1111
01:04:45,420 --> 01:04:51,560
But in general, the convention
is, we write f of x equal to
1112
01:04:51,560 --> 01:04:55,070
that if x is non-zero.
1113
01:04:55,070 --> 01:04:56,730
What are these f's?
1114
01:04:56,730 --> 01:04:58,510
These are called the
coefficients of the
1115
01:04:58,510 --> 01:05:00,050
polynomial.
1116
01:05:00,050 --> 01:05:03,510
And where do they live?
1117
01:05:03,510 --> 01:05:08,160
We need the coefficients to
be in some common field.
1118
01:05:08,160 --> 01:05:11,680
1119
01:05:11,680 --> 01:05:15,010
You've often seen these in the
real or complex field.
1120
01:05:15,010 --> 01:05:17,610
Here they're going to
be in finite fields.
1121
01:05:17,610 --> 01:05:19,310
In particular, very
shortly, they're
1122
01:05:19,310 --> 01:05:22,180
going to be prime fields.
1123
01:05:22,180 --> 01:05:26,480
But in general, we'll just
say these f's have
1124
01:05:26,480 --> 01:05:29,290
to be in some field.
1125
01:05:29,290 --> 01:05:37,460
So we're talking about a
polynomial over F where F is
1126
01:05:37,460 --> 01:05:38,590
some field.
1127
01:05:38,590 --> 01:05:40,640
So there's always some
underlying field if there
1128
01:05:40,640 --> 01:05:42,730
isn't a vector space.
1129
01:05:42,730 --> 01:05:46,970
Some similarities between
this and vector spaces.
1130
01:05:46,970 --> 01:05:54,250
And we usually adopt the
convention that Fm is not
1131
01:05:54,250 --> 01:05:55,500
equal to 0.
1132
01:05:55,500 --> 01:05:57,770
1133
01:05:57,770 --> 01:05:58,090
All right?
1134
01:05:58,090 --> 01:06:01,110
So we only write the polynomial
out to its last
1135
01:06:01,110 --> 01:06:03,160
non-zero coefficient.
1136
01:06:03,160 --> 01:06:09,520
In general, this could go up
to an arbitrary degree, but
1137
01:06:09,520 --> 01:06:14,240
well, a polynomial, by
definition has a finite
1138
01:06:14,240 --> 01:06:18,120
degree, which means it has
a finite m for which the
1139
01:06:18,120 --> 01:06:21,590
polynomial can be written
in this way.
1140
01:06:21,590 --> 01:06:26,710
And if the Fm is the highest
non-zero coefficient, then we
1141
01:06:26,710 --> 01:06:31,340
say the degree of f of x is m.
1142
01:06:31,340 --> 01:06:35,450
1143
01:06:35,450 --> 01:06:41,120
So all polynomials have a finite
degrees, except for 1.
1144
01:06:41,120 --> 01:06:44,820
There is the zero polynomial,
which we have
1145
01:06:44,820 --> 01:06:48,090
to account for somehow.
1146
01:06:48,090 --> 01:06:51,620
And here we'll just call
it f of x equals 0.
1147
01:06:51,620 --> 01:06:54,270
1148
01:06:54,270 --> 01:06:56,910
Informally, it's a polynomial,
all of those
1149
01:06:56,910 --> 01:06:59,560
coefficients are 0.
1150
01:06:59,560 --> 01:07:02,600
But we'll just define it
by its properties.
1151
01:07:02,600 --> 01:07:06,280
Zero polynomial plus any other
polynomial is equal to the
1152
01:07:06,280 --> 01:07:10,370
identity under addition
for the polynomials?
1153
01:07:10,370 --> 01:07:12,710
What's the degree of the
zero polynomial?
1154
01:07:12,710 --> 01:07:17,820
1155
01:07:17,820 --> 01:07:22,060
Anyone have a definition for
the degree of the zero
1156
01:07:22,060 --> 01:07:23,150
polynomial?
1157
01:07:23,150 --> 01:07:24,350
Is this well-defined?
1158
01:07:24,350 --> 01:07:25,600
Undefined?
1159
01:07:25,600 --> 01:07:31,320
1160
01:07:31,320 --> 01:07:32,030
OK.
1161
01:07:32,030 --> 01:07:34,090
Well, I'll suggest to you
that it should be
1162
01:07:34,090 --> 01:07:35,400
defined as minus infinity.
1163
01:07:35,400 --> 01:07:41,970
1164
01:07:41,970 --> 01:07:44,630
This actually makes a lot of
things come out nicely, but it
1165
01:07:44,630 --> 01:07:48,990
is on the other hand, you
don't have to do this.
1166
01:07:48,990 --> 01:07:50,770
If you like, you can define
the degree of
1167
01:07:50,770 --> 01:07:53,270
0 to be minus infinity.
1168
01:07:53,270 --> 01:07:54,520
It's just a convention.
1169
01:07:54,520 --> 01:07:58,450
1170
01:07:58,450 --> 01:07:58,650
OK.
1171
01:07:58,650 --> 01:08:13,520
So the set of all polynomials
over F0.
1172
01:08:13,520 --> 01:08:14,770
What's x here?
1173
01:08:14,770 --> 01:08:19,819
1174
01:08:19,819 --> 01:08:22,250
I've got this thing x.
1175
01:08:22,250 --> 01:08:23,760
What should I think
of this as being?
1176
01:08:23,760 --> 01:08:26,880
Is this an element of a field,
or is it something else?
1177
01:08:26,880 --> 01:08:32,420
1178
01:08:32,420 --> 01:08:35,800
In math, it's usually called
an indeterminate.
1179
01:08:35,800 --> 01:08:37,180
It's just a placeholder.
1180
01:08:37,180 --> 01:08:40,930
It's something else we stick
in order to define the
1181
01:08:40,930 --> 01:08:41,285
polynomial.
1182
01:08:41,285 --> 01:08:45,282
It doesn't have a value,
in principle.
1183
01:08:45,282 --> 01:08:52,960
A comment is made in the notes
that with real and complex
1184
01:08:52,960 --> 01:08:56,939
polynomials, you often think
of x as being a real or
1185
01:08:56,939 --> 01:08:57,960
complex number.
1186
01:08:57,960 --> 01:09:00,890
In other words, you evaluate the
polynomial at some alpha
1187
01:09:00,890 --> 01:09:03,810
in the real or the complex field
by just substituting
1188
01:09:03,810 --> 01:09:05,800
alpha for x.
1189
01:09:05,800 --> 01:09:15,729
And in fact, two polynomials are
equal if they evaluate to
1190
01:09:15,729 --> 01:09:18,479
the same value for all the
alphas and they're
1191
01:09:18,479 --> 01:09:22,220
unequal if not true.
1192
01:09:22,220 --> 01:09:24,760
When we get to finite fields,
it's important this be an
1193
01:09:24,760 --> 01:09:26,180
indeterminate.
1194
01:09:26,180 --> 01:09:32,120
Because consider x and x squared
as polynomials over
1195
01:09:32,120 --> 01:09:36,939
the binary field F2.
1196
01:09:36,939 --> 01:09:40,120
What are the values of these?
1197
01:09:40,120 --> 01:09:48,029
We'll call this F1 of x equals
x, F2 of x equals x squared.
1198
01:09:48,029 --> 01:09:56,320
Then F1 of 0 is 0 and
F1 of 1 is 1.
1199
01:09:56,320 --> 01:09:56,670
Right?
1200
01:09:56,670 --> 01:09:59,950
If I evaluate these at field
elements, the two field
1201
01:09:59,950 --> 01:10:02,060
elements, I get 0 and 1.
1202
01:10:02,060 --> 01:10:10,260
F2 of 0 is equal to 0, and
F2 of 1 is equal to 1.
1203
01:10:10,260 --> 01:10:14,550
But these are not the same
polynomial, all right?
1204
01:10:14,550 --> 01:10:20,880
So x is not to be considered
as a field element.
1205
01:10:20,880 --> 01:10:26,180
It's to be considered just as
a placeholder, a way of
1206
01:10:26,180 --> 01:10:27,710
holding up these polynomials.
1207
01:10:27,710 --> 01:10:31,485
It's actually most important
in multiplication.
1208
01:10:31,485 --> 01:10:34,835
But we gather common
terms in x.
1209
01:10:34,835 --> 01:10:36,950
This is the multiplication
rule.
1210
01:10:36,950 --> 01:10:38,840
But it's just something we
introduce to define the
1211
01:10:38,840 --> 01:10:41,170
polynomial.
1212
01:10:41,170 --> 01:10:41,510
All right.
1213
01:10:41,510 --> 01:10:47,870
So the set of all polynomials
over F in x, or in x over F,
1214
01:10:47,870 --> 01:10:53,800
is simply written as F
square brackets of x.
1215
01:10:53,800 --> 01:10:55,570
That's the convention.
1216
01:10:55,570 --> 01:10:59,320
So that's what I will write
when I mean that.
1217
01:10:59,320 --> 01:11:05,640
And it includes all sequences
like this of finite degree,
1218
01:11:05,640 --> 01:11:07,280
starting at 0, ending
somewhere.
1219
01:11:07,280 --> 01:11:09,870
1220
01:11:09,870 --> 01:11:12,710
And also the zero polynomial.
1221
01:11:12,710 --> 01:11:18,560
1222
01:11:18,560 --> 01:11:19,865
How do you add polynomials?
1223
01:11:19,865 --> 01:11:22,440
1224
01:11:22,440 --> 01:11:26,970
Let's talk about the arithmetic
properties of
1225
01:11:26,970 --> 01:11:27,730
polynomials.
1226
01:11:27,730 --> 01:11:28,780
You know how to do this.
1227
01:11:28,780 --> 01:11:36,230
If you have F0 plus F1 plus F2
and so forth, you have some
1228
01:11:36,230 --> 01:11:40,320
other polynomial doesn't have
to be the same degree --
1229
01:11:40,320 --> 01:11:44,720
G of x is G0 plus G1
x, up there --
1230
01:11:44,720 --> 01:11:47,150
how do you add these together?
1231
01:11:47,150 --> 01:11:49,250
Component-wise.
1232
01:11:49,250 --> 01:11:57,850
Sum is F0 plus G0 plus
F1 plus G1 x plus 2x
1233
01:11:57,850 --> 01:12:00,500
squared and so forth.
1234
01:12:00,500 --> 01:12:03,170
That's an example.
1235
01:12:03,170 --> 01:12:08,280
So you basically insert dummy
0's out here above the highest
1236
01:12:08,280 --> 01:12:11,610
degree term in G. You add
them up component-wise.
1237
01:12:11,610 --> 01:12:13,400
The addition operation
is where?
1238
01:12:13,400 --> 01:12:16,080
In this field, you have addition
operation in that
1239
01:12:16,080 --> 01:12:17,970
failed field, so you
can do this.
1240
01:12:17,970 --> 01:12:21,150
And you get some result which
is clearly itself a
1241
01:12:21,150 --> 01:12:22,580
polynomial.
1242
01:12:22,580 --> 01:12:25,680
If all the coefficients are
0, you declare that
1243
01:12:25,680 --> 01:12:27,260
the result is 0.
1244
01:12:27,260 --> 01:12:30,360
Otherwise the result
has some degree.
1245
01:12:30,360 --> 01:12:33,320
If you add two polynomials with
different degree, the
1246
01:12:33,320 --> 01:12:35,400
degree of the resulting
polynomial is going to be the
1247
01:12:35,400 --> 01:12:36,750
higher degree.
1248
01:12:36,750 --> 01:12:39,840
If they have the same degree,
you could get cancellation in
1249
01:12:39,840 --> 01:12:42,960
the highest order term, and get
a result which is of lower
1250
01:12:42,960 --> 01:12:46,530
degree, all the way down to 0.
1251
01:12:46,530 --> 01:12:47,100
All right?
1252
01:12:47,100 --> 01:12:55,730
So addition is component-wise
the degree of the result is
1253
01:12:55,730 --> 01:13:02,890
less than or equal to the max
degree of the components.
1254
01:13:02,890 --> 01:13:04,445
So we do addition.
1255
01:13:04,445 --> 01:13:05,845
How do we do multiplication?
1256
01:13:05,845 --> 01:13:09,130
1257
01:13:09,130 --> 01:13:11,960
You all know how to do
polynomial multiplication.
1258
01:13:11,960 --> 01:13:14,550
1259
01:13:14,550 --> 01:13:16,300
Example.
1260
01:13:16,300 --> 01:13:24,360
F0 plus F1 of x times
G0 plus G1 of x.
1261
01:13:24,360 --> 01:13:25,170
What do you do?
1262
01:13:25,170 --> 01:13:27,480
You just multiply it
out term by term.
1263
01:13:27,480 --> 01:13:43,530
F0 G0 plus F1 G0 x plus F0 G1
x plus F1 G1 x squared.
1264
01:13:43,530 --> 01:13:45,300
You can combine these
two together.
1265
01:13:45,300 --> 01:13:48,620
1266
01:13:48,620 --> 01:13:51,575
And that's your answer, which
clearly is a polynomial.
1267
01:13:51,575 --> 01:13:58,320
1268
01:13:58,320 --> 01:14:01,390
So that's one way of doing it,
is multiply out term by term,
1269
01:14:01,390 --> 01:14:03,710
collect the terms.
1270
01:14:03,710 --> 01:14:07,970
The result of this is that what
you get is a convolution
1271
01:14:07,970 --> 01:14:10,640
for each of the coefficients
in the new polynomial.
1272
01:14:10,640 --> 01:14:15,980
You convolve, just by the
ordinary rules of polynomial
1273
01:14:15,980 --> 01:14:20,840
addition, you can basically turn
this around, you convolve
1274
01:14:20,840 --> 01:14:22,750
it, and you'll get these
coefficients.
1275
01:14:22,750 --> 01:14:26,330
This is written out
in the notes.
1276
01:14:26,330 --> 01:14:30,530
So we know how to do polynomial
multiplication.
1277
01:14:30,530 --> 01:14:34,360
What are some of
its properties?
1278
01:14:34,360 --> 01:14:36,310
How is this defined
again in F?
1279
01:14:36,310 --> 01:14:38,370
We see we're now going to need
the multiplicative of
1280
01:14:38,370 --> 01:14:46,110
properties of our field F.
All of these products and
1281
01:14:46,110 --> 01:14:49,210
ultimately convolutions are
performed in F. That's why we
1282
01:14:49,210 --> 01:14:53,070
did these coefficients to be in
a field, so we can do all
1283
01:14:53,070 --> 01:14:55,860
these things.
1284
01:14:55,860 --> 01:14:57,690
All right.
1285
01:14:57,690 --> 01:14:58,940
What are some of
the properties?
1286
01:14:58,940 --> 01:15:02,930
1287
01:15:02,930 --> 01:15:08,360
What is the degree of the
product of two polynomials?
1288
01:15:08,360 --> 01:15:12,770
It's going to be the sum
of the degrees, right?
1289
01:15:12,770 --> 01:15:16,650
Provided that both the
polynomials are not 0.
1290
01:15:16,650 --> 01:15:20,790
The highest non-zero term is
clearly going to be a term of
1291
01:15:20,790 --> 01:15:26,450
this kind, and it's going to be
a coefficient of x to the
1292
01:15:26,450 --> 01:15:28,440
sum of the degrees.
1293
01:15:28,440 --> 01:15:32,260
And since F1 and G1 are both
non-zero, by the way we write
1294
01:15:32,260 --> 01:15:35,130
polynomials, them this
highest order term
1295
01:15:35,130 --> 01:15:36,760
is going to be non-zero.
1296
01:15:36,760 --> 01:15:42,080
But we also have to basically
have another rule that 0 times
1297
01:15:42,080 --> 01:15:45,720
f of x is equal to 0.
1298
01:15:45,720 --> 01:15:49,824
So that's the way we
multiply by 0.
1299
01:15:49,824 --> 01:15:53,560
And how does the degree formula
work in this case?
1300
01:15:53,560 --> 01:15:55,360
Well, this is why I defined
the degree of
1301
01:15:55,360 --> 01:15:57,480
0 to be minus infinity.
1302
01:15:57,480 --> 01:16:05,380
So the degree of the product
0 times f of x is --
1303
01:16:05,380 --> 01:16:09,410
I've defined this to be the
degree of 0 plus the
1304
01:16:09,410 --> 01:16:11,260
degree of f of x.
1305
01:16:11,260 --> 01:16:12,130
This is finite.
1306
01:16:12,130 --> 01:16:13,910
This is minus infinity.
1307
01:16:13,910 --> 01:16:18,370
So the sum is minus infinity,
and so it holds.
1308
01:16:18,370 --> 01:16:18,920
OK?
1309
01:16:18,920 --> 01:16:20,980
That's why we defined
the degree of
1310
01:16:20,980 --> 01:16:22,550
0 to be minus infinity.
1311
01:16:22,550 --> 01:16:26,520
We don't have to, but it's just
so that the sum of the
1312
01:16:26,520 --> 01:16:29,890
degrees formula continues
to work, even if we're
1313
01:16:29,890 --> 01:16:32,140
multiplying by 0.
1314
01:16:32,140 --> 01:16:34,550
Is there a multiplicative
identity for polynomials?
1315
01:16:34,550 --> 01:16:38,170
1316
01:16:38,170 --> 01:16:38,440
Yeah.
1317
01:16:38,440 --> 01:16:41,600
What is the multiplicative
identity?
1318
01:16:41,600 --> 01:16:42,180
1.
1319
01:16:42,180 --> 01:16:42,660
OK.
1320
01:16:42,660 --> 01:16:49,175
So one times f of x is
equal to f of x.
1321
01:16:49,175 --> 01:16:51,690
1322
01:16:51,690 --> 01:16:55,900
So that's one of the properties
of a field.
1323
01:16:55,900 --> 01:16:57,450
Gee.
1324
01:16:57,450 --> 01:17:02,920
The set of all polynomials over
F in x, is this a field?
1325
01:17:02,920 --> 01:17:06,710
Let's go back and check
our field axioms.
1326
01:17:06,710 --> 01:17:10,245
Under addition, does f of
x form an abelian group?
1327
01:17:10,245 --> 01:17:13,640
1328
01:17:13,640 --> 01:17:14,740
Does it have the
group property?
1329
01:17:14,740 --> 01:17:16,900
If we add two polynomials, do
we get another polynomial?
1330
01:17:16,900 --> 01:17:19,460
1331
01:17:19,460 --> 01:17:24,590
Do we have cancellation, if F1
plus F2 equals F1 plus F3, is
1332
01:17:24,590 --> 01:17:26,320
it necessarily true
that F2 equals F3?
1333
01:17:26,320 --> 01:17:29,440
1334
01:17:29,440 --> 01:17:30,390
Yeah, it is.
1335
01:17:30,390 --> 01:17:33,360
In fact, this looks very
much under addition.
1336
01:17:33,360 --> 01:17:36,480
1337
01:17:36,480 --> 01:17:39,815
These look like vectors.
1338
01:17:39,815 --> 01:17:42,960
If we confine ourselves to the
set of all polynomials of
1339
01:17:42,960 --> 01:17:46,660
degree m or less,
we can add them.
1340
01:17:46,660 --> 01:17:54,400
And it looks very much like the
vector space f to the m.
1341
01:17:54,400 --> 01:17:59,130
Set of all polynomials of degree
m or less corresponds
1342
01:17:59,130 --> 01:18:04,140
to the vector space Fm, which
does have the group property
1343
01:18:04,140 --> 01:18:07,450
under addition.
1344
01:18:07,450 --> 01:18:12,732
So in fact, for f of x, yes.
1345
01:18:12,732 --> 01:18:16,830
It is an abelian group under
addition with identity being
1346
01:18:16,830 --> 01:18:18,080
the 0 polynomial.
1347
01:18:18,080 --> 01:18:21,530
1348
01:18:21,530 --> 01:18:26,030
And all of f of x is an infinite
abelian group.
1349
01:18:26,030 --> 01:18:29,150
If we just take polynomials of
degree m or less and restrict
1350
01:18:29,150 --> 01:18:32,060
Fp to be a finite group, then
there are only p to the m
1351
01:18:32,060 --> 01:18:37,230
elements, and we would have
a finite abelian group.
1352
01:18:37,230 --> 01:18:38,370
OK.
1353
01:18:38,370 --> 01:18:38,950
Well.
1354
01:18:38,950 --> 01:18:44,405
Are the polynomials an abelian
group under multiplication?
1355
01:18:44,405 --> 01:18:47,350
1356
01:18:47,350 --> 01:18:49,780
It has an identity.
1357
01:18:49,780 --> 01:18:54,020
It has all the arithmetic
properties you might expect.
1358
01:18:54,020 --> 01:18:55,170
It's commutative.
1359
01:18:55,170 --> 01:18:59,130
f of x times g of x is equal
to g of x times f of x.
1360
01:18:59,130 --> 01:19:00,380
So it's abelian.
1361
01:19:00,380 --> 01:19:02,610
1362
01:19:02,610 --> 01:19:05,560
It has cancellation.
1363
01:19:05,560 --> 01:19:08,840
f of x g of x is equal
to f of x h of x.
1364
01:19:08,840 --> 01:19:12,520
That's true if and only if g
of x is equal to h of x.
1365
01:19:12,520 --> 01:19:16,040
1366
01:19:16,040 --> 01:19:18,520
Is it missing anything?
1367
01:19:18,520 --> 01:19:19,710
Inverse, right?
1368
01:19:19,710 --> 01:19:21,450
Very good.
1369
01:19:21,450 --> 01:19:22,700
Just like the integers.
1370
01:19:22,700 --> 01:19:24,965
Not all the polynomials
have inverses.
1371
01:19:24,965 --> 01:19:28,610
1372
01:19:28,610 --> 01:19:32,808
Which are the polynomials
that do have inverses?
1373
01:19:32,808 --> 01:19:34,058
AUDIENCE: [INAUDIBLE]
1374
01:19:34,058 --> 01:19:37,310
1375
01:19:37,310 --> 01:19:37,750
PROFESSOR: No.
1376
01:19:37,750 --> 01:19:41,320
There are more than that.
1377
01:19:41,320 --> 01:19:46,983
So now we're getting into
polynomial factorization.
1378
01:19:46,983 --> 01:19:51,430
1379
01:19:51,430 --> 01:19:57,100
And the particular topic
is units, which are the
1380
01:19:57,100 --> 01:19:58,350
invertible polynomials.
1381
01:19:58,350 --> 01:20:06,870
1382
01:20:06,870 --> 01:20:08,120
And what are they?
1383
01:20:08,120 --> 01:20:10,294
1384
01:20:10,294 --> 01:20:11,970
Does the 0 polynomial
have an inverse?
1385
01:20:11,970 --> 01:20:20,960
1386
01:20:20,960 --> 01:20:22,210
We're a little unsure, are we?
1387
01:20:22,210 --> 01:20:24,836
1388
01:20:24,836 --> 01:20:28,045
What could it possibly be?
1389
01:20:28,045 --> 01:20:30,230
If it had an inverse,
this would mean 0
1390
01:20:30,230 --> 01:20:34,510
times f of x is --
1391
01:20:34,510 --> 01:20:39,450
well, 0 times f of x would have
to be a one-to-one map to
1392
01:20:39,450 --> 01:20:42,590
all of f of x.
1393
01:20:42,590 --> 01:20:44,280
But it isn't.
1394
01:20:44,280 --> 01:20:46,450
It simply maps to 0.
1395
01:20:46,450 --> 01:20:50,270
Doesn't have an inverse.
1396
01:20:50,270 --> 01:20:55,520
What about the non-zero
polynomials
1397
01:20:55,520 --> 01:20:56,520
that have degree 0?
1398
01:20:56,520 --> 01:21:06,270
In other words, the degree
0 polynomial, is simply
1399
01:21:06,270 --> 01:21:10,650
something that looks like this.
fx equals f0, where f0
1400
01:21:10,650 --> 01:21:11,900
is not equal to 0.
1401
01:21:11,900 --> 01:21:16,400
1402
01:21:16,400 --> 01:21:19,386
Is that invertible?
1403
01:21:19,386 --> 01:21:22,600
Yeah, because f0 is in the
field, and it has an inverse.
1404
01:21:22,600 --> 01:21:29,930
So this has inverse 1 over f of
x, if you like, equals just
1405
01:21:29,930 --> 01:21:31,340
f0 minus 1.
1406
01:21:31,340 --> 01:21:32,950
1 over f0.
1407
01:21:32,950 --> 01:21:38,880
1408
01:21:38,880 --> 01:21:40,810
By the rules, you multiply
these two
1409
01:21:40,810 --> 01:21:42,085
things, and you get 1.
1410
01:21:42,085 --> 01:21:48,630
1411
01:21:48,630 --> 01:21:49,810
OK.
1412
01:21:49,810 --> 01:21:53,280
So the units in this --
1413
01:21:53,280 --> 01:21:54,710
well, I'm sorry.
1414
01:21:54,710 --> 01:21:58,460
Take a degree 1 or a higher
polynomial --
1415
01:21:58,460 --> 01:21:59,710
does that have an inverse?
1416
01:21:59,710 --> 01:22:03,640
1417
01:22:03,640 --> 01:22:06,535
Let's suppose we take a
degree 1 polynomial --
1418
01:22:06,535 --> 01:22:10,680
1419
01:22:10,680 --> 01:22:16,000
say, F0 plus F1 x --
1420
01:22:16,000 --> 01:22:18,010
and what I want to find
is its inverse.
1421
01:22:18,010 --> 01:22:21,160
1422
01:22:21,160 --> 01:22:24,010
Let's call it g of x.
1423
01:22:24,010 --> 01:22:26,610
Is it possible to find a g of
x such that the product of
1424
01:22:26,610 --> 01:22:27,860
these two is 1?
1425
01:22:27,860 --> 01:22:35,400
1426
01:22:35,400 --> 01:22:39,170
Clearly not, because by our
degree rule, what is the
1427
01:22:39,170 --> 01:22:41,850
degree of this product
going to be?
1428
01:22:41,850 --> 01:22:45,720
The degree is going to be the
sum of this degree plus this
1429
01:22:45,720 --> 01:22:50,200
degree, the degree of f plus
the degree of g, which is
1430
01:22:50,200 --> 01:22:53,850
going to have to
be at least 1.
1431
01:22:53,850 --> 01:22:56,780
Provided that g of x is not 0,
but clearly g of x is not the
1432
01:22:56,780 --> 01:22:59,600
solution we're looking
for here, either.
1433
01:22:59,600 --> 01:23:08,910
So this can't be true, and the
invertible polynomials are the
1434
01:23:08,910 --> 01:23:17,850
degree 0 polynomials, which
means they're basically the
1435
01:23:17,850 --> 01:23:24,050
non-zero elements of F. Yes.
1436
01:23:24,050 --> 01:23:25,150
Considered as polynomials.
1437
01:23:25,150 --> 01:23:26,785
AUDIENCE: And how are the
[UNINTELLIGIBLE PHRASE]?
1438
01:23:26,785 --> 01:23:33,000
1439
01:23:33,000 --> 01:23:33,690
PROFESSOR: Yes, indeed.
1440
01:23:33,690 --> 01:23:34,940
AUDIENCE: [INAUDIBLE]
1441
01:23:34,940 --> 01:23:37,920
1442
01:23:37,920 --> 01:23:38,610
PROFESSOR: Ah, no.
1443
01:23:38,610 --> 01:23:43,560
We haven't introduced modulo
polynomials and right.
1444
01:23:43,560 --> 01:23:48,010
1445
01:23:48,010 --> 01:23:53,480
The powers and the indices are
the integers from 0 up to some
1446
01:23:53,480 --> 01:23:54,730
finite number.
1447
01:23:54,730 --> 01:23:57,170
1448
01:23:57,170 --> 01:23:58,160
OK.
1449
01:23:58,160 --> 01:23:59,790
It's time to quit.
1450
01:23:59,790 --> 01:24:02,750
We'll finish this.
1451
01:24:02,750 --> 01:24:06,400
We'll, I believe, to be able
to certainly finish chapter
1452
01:24:06,400 --> 01:24:10,140
seven, maybe get a little bit
into chapter eight, next time,
1453
01:24:10,140 --> 01:24:12,945
on Wednesday.
1454
01:24:12,945 --> 01:24:14,195
And we'll see you then.
1455
01:24:14,195 --> 01:24:46,548