WEBVTT
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PROFESSOR: So we're slightly
into chapter seven, which is
00:00:05.940 --> 00:00:08.029
the algebra chapter.
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We're talking about a number of
algebraic objects, starting
00:00:12.600 --> 00:00:16.070
with integers and groups
and fields.
00:00:16.070 --> 00:00:16.970
Polynomials.
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Our objective in this chapter
is simply to get to finite
00:00:21.700 --> 00:00:27.370
fields so that you have some
sense what they are, how they
00:00:27.370 --> 00:00:30.780
can be constructed, what their
parameters are, how you can
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operate with them by addition,
multiplication, so forth, as
00:00:34.470 --> 00:00:36.630
you would expect in a field.
00:00:36.630 --> 00:00:39.440
Subtraction, division.
00:00:39.440 --> 00:00:42.380
And that's really all
we're aiming to do.
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I'm trying to give you a short
course in algebra, really, in
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two lectures or fewer.
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And clearly I'm going to
miss a lot of things.
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In particular, I'm not going
to cover even everything
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that's in chapter seven, which
itself is a highly compressed
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introduction to finite fields.
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I'm trying to do this while
remaining faithful to the
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philosophy of this course and
other courses at MIT, which is
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that you should really prove
everything and show why things
00:01:12.880 --> 00:01:15.810
are true, and not simply
make assertions.
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So it's a little tough and it
forces me to go a little fast,
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but I hope that you can keep
up, and especially with the
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assistance of the notes or the
many possible other things you
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could read on this subject,
which are listed in the notes,
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you'll be able to keep up.
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And some of you, of course, have
seen this in other places
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in more extended form.
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Now, this will get us in a
position to start to talk
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about Reed-Solomon codes, which
are the single major
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accomplishment of the field of
algebraic coding theory.
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Certainly for getting to
capacity on the additive white
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Gaussian noise channel and for
lots of other things, they're
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an extremely useful and widely
implemented class of codes.
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And we'll be able to maybe just
get to the beginning of
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that by Wednesday.
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And then I won't be here again
next week, but fortunately we
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have an expert on campus who is
far more expert than I in
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Reed-Solomon codes, their
decoding algorithms, who has
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agreed to talk for two lectures,
maybe one more
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focused on Reed-Solomon codes
and one, I hope, on his whole
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philosophy of life.
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Perhaps.
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I don't know how it's going to
come out, but I'll see it on
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TV when I get back.
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Anyway, Ralf Koetter will be
lecturer for Monday and
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Wednesday next week.
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I think you'll enjoy
the change of pace.
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OK.
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So where are we in
chapter seven?
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We're not very far.
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We're talking about these
various algebraic objects.
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We've started with integers
just to get you
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into the feel of it.
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We mainly talked about integer
factorization, the Euclidean
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division algorithm, things that
you've known for a very
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long time, basically here
because A, we're going to be
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using integers as we go along,
and their factorization
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properties, B, it's a model for
polynomials, which behave
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very much the same way as
integers because they're both
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principal ideal domains.
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In particular, we looked at the
integers mod n with the
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rules of mod n arithmetic, which
we're going to call Zn.
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This is simply 0 through n
minus 1 with the mod n
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arithmetic rules.
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And then we went on to groups.
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We first gave the standard
axioms for groups, and then I
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gave you an alternative set of
axioms which focused on this
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permutation property.
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If you add, I'm calling the
group operation addition,
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because essentially all the
groups we talk about are going
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to be abelian --
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if we add a group element
to the group,
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what do we get back?
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We get the whole group again.
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It's permuted, it's the
entire group, it's
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in a different order.
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All right?
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And with this and the identity,
this plus the
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identity and the associativity
axiom are also a sufficient
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set of axioms for the group.
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And I think this is really the
most useful thing to think
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about with a group.
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We've also called it the
group property when we
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talked about codes.
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You know, if you add the code
word to all the elements in
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the code, you get
the code back.
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And you saw how useful that
was for seeing certain
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symmetry properties of minor
linear block codes.
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So we even talked about cyclic
groups, specifically finite
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cyclic groups.
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And we showed that all
them basically are
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isomorphic to z mod n.
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A cyclic group is defined
by a single generator.
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If we identify that generator
with one, G plus G is
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identified with 2, and so forth,
then we get an addition
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table which is exactly the same
addition table as Zn.
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And that's what we mean when
we say two groups are
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isomorphic.
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So all finite cyclic groups
look like Zn.
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You can think of them as
being images of Zn.
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All right?
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It's the only one you
need to know about.
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OK.
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So that's where we are any
questions on this material?
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Pretty easy, I think.
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Terribly easy if you've ever
seen any of this before.
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Probably takes a little
absorbing if you haven't.
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OK.
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Now the next natural subject
to talk about is subgroups.
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And what is a subgroup?
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A subgroup simply a subset of
elements in the group which,
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together with the group
operation already specified in
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G, which we're calling circle
plus, is itself a group.
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What does that mean?
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Well, associativity comes for
free, because we already have
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that property for circle plus
in G. Obviously H has to
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include the identity in order
to satisfy the group axioms.
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And finally, the third group
axiom is this permutation or
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group property that if we add
any element of the subgroup to
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itself, we have to stay within
the subgroup and ultimately
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generate the whole subgroup.
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That means that subtraction,
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cancellation hold in the subgroup.
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All right?
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So that's clear.
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What's an example
of a subgroup?
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If we have --
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we talked about Z10
as the group.
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What would be a subgroup?
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Anybody?
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AUDIENCE: [INAUDIBLE]
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PROFESSOR: 0 to 4.
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OK.
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So you're proposing that H is
the elements 0, 1, 2, 3, 4 out
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of G. Does that work?
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This doesn't include 0.
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But suppose I add 3 and 4?
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AUDIENCE: I assume that you
have modules for a reason.
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PROFESSOR: No.
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In this group, the group
operations modulo 10 addition.
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Subgroup has to have the
same operation as the
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group it came from.
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What you've got here is you've
already got, in essence, a
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quotient group.
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Or at least that's where
you're headed.
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So this is not a subgroup.
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Thank you for the suggestion.
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Fails.
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Anyone else?
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What?
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AUDIENCE: 0,1?
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PROFESSOR: 0,1.
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OK.
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Let's try that.
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And it contains the identity
0 plus 1 is
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certainly in the group.
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How about 1 plus 1?
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It's not in the group.
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0 and 5?
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That sounds more promising.
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Now, 0 plus -- what's the
addition table of this?
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0, 5, 0, 5, 0, 5.
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What's 5 plus 5?
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It's 10.
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But mod-10 , that's 0.
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So it seems we do
have a group.
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And in fact, this is a finite
cyclic group generated by 5,
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and has two elements, so
it's isomorphic to Z2.
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In other words, the addition
table looks just like the
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addition table of Z2
with a relabeling.
00:09:05.500 --> 00:09:06.860
OK.
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Any other subgroups of Z10?
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The even integers.
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There.
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Now we're really smoking.
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H equals 0,2,4,6,8.
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Those are all the even
integers in Z10.
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And again, evens plus
evens equal evens.
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So we get a group of five
elements, satisfies is the
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group property, and it's
isomorphic to what?
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Z5 --
00:09:39.845 --> 00:09:41.310
yeah.
00:09:41.310 --> 00:09:44.490
This is obviously the same group
is 0, 1, 2, 3, 4, just
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doubling everything.
00:09:48.700 --> 00:09:49.720
Operates the same way.
00:09:49.720 --> 00:09:50.970
So it's change of labels.
00:09:53.350 --> 00:09:53.460
OK.
00:09:53.460 --> 00:09:55.485
So there are some examples
of subgroups.
00:09:59.030 --> 00:10:03.700
Let's take another
good example.
00:10:03.700 --> 00:10:05.680
Let's just take the set
of all integers.
00:10:05.680 --> 00:10:07.780
That's an infinite group.
00:10:07.780 --> 00:10:12.400
Mathematicians call it cyclic,
even though it doesn't cycle.
00:10:12.400 --> 00:10:13.800
What's a subgroup of that?
00:10:23.530 --> 00:10:24.460
AUDIENCE: Even integers.
00:10:24.460 --> 00:10:25.450
PROFESSOR: All even integers!
00:10:25.450 --> 00:10:28.140
Very good.
00:10:28.140 --> 00:10:28.420
Check.
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Does that include 0?
00:10:29.670 --> 00:10:30.370
Yes.
00:10:30.370 --> 00:10:31.910
Does it have the group
property, even
00:10:31.910 --> 00:10:34.240
plus even is even?
00:10:34.240 --> 00:10:36.670
Clearly subtraction holds,
and so forth.
00:10:36.670 --> 00:10:39.560
So this is the subgroup.
00:10:42.190 --> 00:10:44.860
Interestingly, there's a
one-to-one correspondence
00:10:44.860 --> 00:10:47.440
between Z and 2Z, so the
subgroup is as big as the
00:10:47.440 --> 00:10:50.420
group itself.
00:10:50.420 --> 00:10:56.940
You get into the whole issue
of transfinite numbers.
00:10:56.940 --> 00:10:58.570
But that's not where
we're headed here.
00:11:01.150 --> 00:11:03.030
OK.
00:11:03.030 --> 00:11:05.390
We'll just keep that
in mind for now.
00:11:05.390 --> 00:11:13.350
Obviously 3Z, 4Z, 5Z, and all
the multiples of n are going
00:11:13.350 --> 00:11:17.390
to be a subgroup of the integers
for any integer n.
00:11:17.390 --> 00:11:21.930
So more generally, we could
take H equals nZ.
00:11:34.400 --> 00:11:35.650
What's a coset?
00:11:38.100 --> 00:11:42.450
Coset is also, in the abelian
case, called a
00:11:42.450 --> 00:11:45.306
translate of a subgroup.
00:11:50.820 --> 00:11:59.820
A coset, for instance, is in the
form H plus G for some G
00:11:59.820 --> 00:12:01.070
in the group, not
in the subgroup.
00:12:04.670 --> 00:12:09.730
Now if G is in the subgroup,
we get nothing.
00:12:09.730 --> 00:12:16.970
The coset is just H again by
the group property of H. So
00:12:16.970 --> 00:12:25.700
the interesting cases are where
G is not in H. Let me
00:12:25.700 --> 00:12:27.820
give you some examples.
00:12:27.820 --> 00:12:41.290
Let's take G equals Z10 and
H equals 0,2,4,6,8.
00:12:41.290 --> 00:12:47.720
You might call that 2Z10 It's
the set of all elements which
00:12:47.720 --> 00:12:52.050
are twice the elements in Z10.
00:12:52.050 --> 00:12:52.500
OK.
00:12:52.500 --> 00:12:54.600
What is a coset?
00:12:54.600 --> 00:12:58.180
If I add any element in
this group to H --
00:12:58.180 --> 00:13:01.780
let's add one, for instance.
00:13:01.780 --> 00:13:13.585
So H plus 1 consists of the
elements 1,3,5,7,9.
00:13:13.585 --> 00:13:14.835
Well, that's interesting.
00:13:17.000 --> 00:13:25.130
That seems to exhaust all
of the elements of Z10.
00:13:25.130 --> 00:13:33.370
Let's take the Z equals the Z10,
and H simply equal to 0
00:13:33.370 --> 00:13:35.550
and 5, which we might
call 5Z10.
00:13:39.650 --> 00:13:42.000
And all right.
00:13:42.000 --> 00:13:46.570
Now H plus 0, H plus 5 is
just equal to itself.
00:13:46.570 --> 00:13:50.650
H plus 1 is equal to 1,6.
00:13:50.650 --> 00:13:55.240
H plus 2 is equal to 2,7.
00:13:55.240 --> 00:14:01.040
H plus 3 is equal to 3, 8.
00:14:01.040 --> 00:14:10.770
H plus 4 is equal to 4,9.
00:14:10.770 --> 00:14:18.320
So we begin to see some
properties of cosets here for
00:14:18.320 --> 00:14:22.050
which proofs are given
in the notes.
00:14:22.050 --> 00:14:27.760
First of all is that
two cosets are
00:14:27.760 --> 00:14:31.020
either the same or disjoint.
00:14:31.020 --> 00:14:36.760
H plus 2 is the same as H. H
plus 1 is completely disjoint
00:14:36.760 --> 00:14:42.520
from H. Same over here.
00:14:42.520 --> 00:14:49.450
If I had H plus 5, that would
be the same as H. H plus 6
00:14:49.450 --> 00:14:52.360
would be completely disjoint
from H and would be the same
00:14:52.360 --> 00:14:54.300
as H plus 1.
00:14:54.300 --> 00:14:58.650
In fact, I can take any of the
elements of a coset as its
00:14:58.650 --> 00:15:01.630
representative, and I'm going
to get the same coset, take
00:15:01.630 --> 00:15:04.040
any element outside the coset,
and I'll get a completely
00:15:04.040 --> 00:15:05.010
distinct coset.
00:15:05.010 --> 00:15:07.370
This follows just very
easily from the
00:15:07.370 --> 00:15:10.610
cancellation property.
00:15:10.610 --> 00:15:11.900
All right?
00:15:11.900 --> 00:15:24.170
So the cosets, the distinct
cosets, form a disjoint
00:15:24.170 --> 00:15:36.780
partition of G. We certainly
have a coset that contains
00:15:36.780 --> 00:15:40.210
every element of G. Just take H
plus that element G. That's
00:15:40.210 --> 00:15:43.870
going to contain G because
H contains 0.
00:15:43.870 --> 00:15:49.990
So there is a coset that
contains every element of G.
00:15:49.990 --> 00:15:54.030
Any two cosets, distinct cosets,
are disjoint, so
00:15:54.030 --> 00:15:56.350
that's what we mean by
a disjoint partition.
00:15:56.350 --> 00:16:00.470
We list all the elements
of G in this way.
00:16:03.130 --> 00:16:09.700
And in the finite case, that
gives us a early famous
00:16:09.700 --> 00:16:14.220
theorem attributed
to Lagrange.
00:16:17.440 --> 00:16:18.600
What does that mean?
00:16:18.600 --> 00:16:35.800
This means that H has to divide
G. If G has size 1,10
00:16:35.800 --> 00:16:40.970
and all the cosets have the same
size, by the way, again
00:16:40.970 --> 00:16:43.450
by the cancellation
property --
00:16:43.450 --> 00:16:48.040
this means that G has to consist
of an integer number
00:16:48.040 --> 00:16:53.410
of cosets, all of which have
the size of H. OK?
00:16:53.410 --> 00:16:57.410
And therefore some integer times
H is equal to H, which
00:16:57.410 --> 00:17:01.240
is the same thing as
H divides G. OK?
00:17:13.170 --> 00:17:18.210
If G is a finite group, I mean
by this kind of determinant
00:17:18.210 --> 00:17:22.619
notation, the size of H. The
size of H divides the size of
00:17:22.619 --> 00:17:26.890
G. Or more elegantly, the
cardinality of H divides the
00:17:26.890 --> 00:17:29.350
cardinality of G. But
why say cardinality
00:17:29.350 --> 00:17:32.540
when you can say size?
00:17:32.540 --> 00:17:33.792
I shouldn't have put it here.
00:17:36.990 --> 00:17:41.210
Where H is any subgroup, any
subgroup, of course, that's
00:17:41.210 --> 00:17:44.850
finite is itself
a finite group.
00:17:44.850 --> 00:17:47.630
So that's going to turn out to
be quite a powerful theorem.
00:17:47.630 --> 00:17:51.770
And it just follows this
little exercise.
00:17:51.770 --> 00:17:54.500
And we see it's satisfied,
certainly,
00:17:54.500 --> 00:17:56.680
by these two examples.
00:17:59.270 --> 00:18:06.970
In fact, it's pretty easy to see
that the subgroups of Zm
00:18:06.970 --> 00:18:10.720
are going to correspond to the
divisors of m in the same way.
00:18:10.720 --> 00:18:14.050
We're going to get a subgroup
for every divisor of m, and in
00:18:14.050 --> 00:18:16.375
the case of cyclic groups,
this is the way it's
00:18:16.375 --> 00:18:17.310
going to come out.
00:18:17.310 --> 00:18:18.800
Just pick any divisor.
00:18:18.800 --> 00:18:24.150
You get a subgroup
isomorphic to Zd.
00:18:24.150 --> 00:18:27.970
Again, this is done with
more care in the notes.
00:18:27.970 --> 00:18:30.100
Suppose we have the infinite
case here.
00:18:30.100 --> 00:18:32.920
Suppose we have Z and 2Z.
00:18:32.920 --> 00:18:36.750
So let me draw that case.
00:18:36.750 --> 00:18:41.150
G equals Z. H equals 2Z.
00:18:43.680 --> 00:18:44.970
And all right.
00:18:44.970 --> 00:18:48.400
What's H?
00:18:48.400 --> 00:18:58.730
H is the set dot dot dot minus
2,0,2,4, dot dot dot.
00:18:58.730 --> 00:19:06.870
So this is H. And what's H plus
1, or in fact, plus any
00:19:06.870 --> 00:19:08.520
odd number?
00:19:08.520 --> 00:19:11.210
It's going to be the
odd integers.
00:19:11.210 --> 00:19:16.120
H equals minus 1,1,3,5,
and so forth.
00:19:18.790 --> 00:19:26.100
So again, this works in the
infinite case, that the
00:19:26.100 --> 00:19:29.850
distinct cosets form a disjoint
partition of an
00:19:29.850 --> 00:19:33.105
infinite group G. But of
course, we don't get
00:19:33.105 --> 00:19:38.040
Lagrange's theorem as a
corollary, because I already
00:19:38.040 --> 00:19:40.250
said, there's a one-to-one
correspondence between Z and
00:19:40.250 --> 00:19:44.360
2Z, paradoxically.
00:19:44.360 --> 00:19:48.160
So the cardinality of Z
and 2Z are the same.
00:19:48.160 --> 00:19:50.130
More elegant language.
00:19:50.130 --> 00:19:53.070
Nonetheless, you see, this
is a useful partition and
00:19:53.070 --> 00:19:56.220
standard partition
into the even
00:19:56.220 --> 00:19:59.030
integers and the odd integers.
00:19:59.030 --> 00:20:04.990
And we could also write this as
2Z and this is 2Z plus 1.
00:20:04.990 --> 00:20:06.930
So we can divide the
integers into even
00:20:06.930 --> 00:20:08.180
integers and odd integers.
00:20:10.640 --> 00:20:11.350
All right.
00:20:11.350 --> 00:20:18.360
Now we can actually add cosets,
subtract cosets.
00:20:18.360 --> 00:20:21.110
In these cases, we can
even multiply cosets.
00:20:21.110 --> 00:20:23.090
But let's just talk
about staying
00:20:23.090 --> 00:20:25.760
within the group operation.
00:20:25.760 --> 00:20:27.600
Given any abelian --
00:20:27.600 --> 00:20:29.870
let's continue to say
G is abelian --
00:20:29.870 --> 00:20:31.670
how would you add two cosets?
00:20:34.340 --> 00:20:42.060
Coset addition is defined
as follows.
00:20:42.060 --> 00:20:44.730
H plus G --
00:20:44.730 --> 00:20:48.290
we want to have some addition
operation, which I'll just
00:20:48.290 --> 00:20:50.860
indicate by plus --
00:20:50.860 --> 00:20:54.810
H plus G prime, what's
going to equal?
00:20:54.810 --> 00:21:04.065
We define that to equal
H plus G plus G prime.
00:21:07.490 --> 00:21:09.110
And that makes sense.
00:21:09.110 --> 00:21:15.830
I mean, if we really write all
this out, we get H plus H plus
00:21:15.830 --> 00:21:18.940
G plus G prime.
00:21:18.940 --> 00:21:21.490
H plus H is just H again.
00:21:21.490 --> 00:21:24.740
So it's sort of a
proof of that.
00:21:24.740 --> 00:21:29.460
If you go through in detail, any
element of this coset plus
00:21:29.460 --> 00:21:32.330
any element of this coset
is going to be an
00:21:32.330 --> 00:21:34.930
element of this coset.
00:21:34.930 --> 00:21:40.380
So this itself is a coset.
00:21:40.380 --> 00:21:42.680
So we now have an addition
table for cosets.
00:21:47.780 --> 00:21:51.340
So in fact, it's easy to show
that the cosets themselves
00:21:51.340 --> 00:21:56.700
form a group called
a quotient group.
00:21:56.700 --> 00:21:57.950
Start over here.
00:22:05.400 --> 00:22:16.073
Cosets of H in G under
coset addition --
00:22:16.073 --> 00:22:17.323
that's going to be Z --
00:22:20.960 --> 00:22:22.600
form a group.
00:22:22.600 --> 00:22:25.370
We just defined coset
addition.
00:22:25.370 --> 00:22:30.310
And it's easy to check that they
themselves form a group
00:22:30.310 --> 00:22:32.980
called the quotient group.
00:22:37.360 --> 00:22:45.450
Usually written G slash H and
pronounced G mod H. And we can
00:22:45.450 --> 00:22:47.570
can mod out anything.
00:22:47.570 --> 00:22:51.190
We do the arithmetic in these
quotient groups by modding out
00:22:51.190 --> 00:22:54.560
any elements of H.
00:22:54.560 --> 00:22:56.030
And let's take an example.
00:22:56.030 --> 00:22:57.280
Here's a good one.
00:23:04.540 --> 00:23:13.980
For example, the cosets
of 2Z in Z, namely,
00:23:13.980 --> 00:23:18.425
2Z and 2Z plus 1.
00:23:24.050 --> 00:23:25.730
Under coset addition.
00:23:25.730 --> 00:23:28.142
What is coset addition here?
00:23:28.142 --> 00:23:36.330
If I add any even integer to
any even integer, I get an
00:23:36.330 --> 00:23:38.510
even integer.
00:23:38.510 --> 00:23:41.455
Any odd to even, I get
an odd integer.
00:23:41.455 --> 00:23:44.880
Odd to odd gives that.
00:23:44.880 --> 00:23:50.370
I mean, odd to even gives that,
and odd to odd gives me
00:23:50.370 --> 00:23:52.550
back evens again.
00:23:52.550 --> 00:23:55.660
So that's the addition table.
00:23:55.660 --> 00:23:59.690
The subgroup itself,
x is the identity.
00:23:59.690 --> 00:24:08.660
This is clearly isomorphic to
Z2 with this addition table.
00:24:12.420 --> 00:24:16.920
In fact, this is a very good way
of constructing the cyclic
00:24:16.920 --> 00:24:20.660
group Z2, or more
generally, Zn.
00:24:20.660 --> 00:24:27.920
So this would be called
Z mod 2Z.
00:24:27.920 --> 00:24:34.850
And it's isomorphic to Z2, or
in general, Z mod nZ is
00:24:34.850 --> 00:24:36.100
isomorphic to Zn.
00:24:38.800 --> 00:24:44.150
A very good way of thinking of
Zn is as residue classes or
00:24:44.150 --> 00:24:45.780
equivalence classes, modulo n.
00:24:51.810 --> 00:24:54.180
The cosets of nZ.
00:24:54.180 --> 00:25:00.880
are nZ itself, nZ plus 1,
nZ plus 2, up to nZ
00:25:00.880 --> 00:25:05.030
plus n minus 1.
00:25:05.030 --> 00:25:08.780
And if you add them together,
they follow the rules of mod n
00:25:08.780 --> 00:25:09.540
arithmetic.
00:25:09.540 --> 00:25:11.970
If you just add the residues
together and
00:25:11.970 --> 00:25:13.500
then reduce mod n.
00:25:13.500 --> 00:25:19.040
So we can think of
Zn as being --
00:25:19.040 --> 00:25:21.770
a coset is an equivalence
class.
00:25:21.770 --> 00:25:24.160
It's all the elements of the
group that are equivalent,
00:25:24.160 --> 00:25:29.830
modular of the subgroup H. Or in
the case of integers, it's
00:25:29.830 --> 00:25:32.460
all integers that are
equivalent modulo
00:25:32.460 --> 00:25:34.870
the subgroup nZ.
00:25:34.870 --> 00:25:39.280
They have the same remainder
after division by n.
00:25:39.280 --> 00:25:40.370
They have the same residue.
00:25:40.370 --> 00:25:42.765
These are all equivalence
class notions.
00:25:42.765 --> 00:25:45.510
And how do you add them?
00:25:45.510 --> 00:25:50.020
You add them in the ordinary
way, and then you take
00:25:50.020 --> 00:25:52.120
everything modulo m.
00:25:52.120 --> 00:25:56.440
In other words, you do
mod-m arithmetic.
00:25:56.440 --> 00:25:57.090
OK.
00:25:57.090 --> 00:26:05.030
So I didn't quite go to quotient
groups in the notes,
00:26:05.030 --> 00:26:08.130
but perhaps I should have.
00:26:08.130 --> 00:26:13.730
Probably I should have, because
this is maybe the most
00:26:13.730 --> 00:26:19.500
powerful idea in group theory,
and certainly closely related
00:26:19.500 --> 00:26:23.150
to this little bit of number
theory that we're doing in the
00:26:23.150 --> 00:26:25.050
integers mod n.
00:26:25.050 --> 00:26:28.600
And of course, it has
vastly greater
00:26:28.600 --> 00:26:32.390
applications than just that.
00:26:32.390 --> 00:26:32.720
OK.
00:26:32.720 --> 00:26:37.410
And you could do the same
thing over here.
00:26:37.410 --> 00:26:42.910
This is basically doing the
same kind of thing.
00:26:45.890 --> 00:26:48.175
But I won't take time
to do that.
00:26:50.710 --> 00:26:55.470
So here's another view of the
integers mod-n that may be
00:26:55.470 --> 00:26:56.830
helpful as we go forward.
00:27:00.590 --> 00:27:01.120
All right.
00:27:01.120 --> 00:27:05.740
I think that's all I want
to say about that.
00:27:05.740 --> 00:27:07.070
Yeah.
00:27:07.070 --> 00:27:07.490
Good.
00:27:07.490 --> 00:27:08.040
All right.
00:27:08.040 --> 00:27:10.025
Here would be a good place
to start on fields.
00:27:18.580 --> 00:27:19.980
OK.
00:27:19.980 --> 00:27:21.230
Fields.
00:27:22.850 --> 00:27:25.100
Obviously very important
in algebra.
00:27:28.160 --> 00:27:30.490
Fields are like groups,
only more so.
00:27:30.490 --> 00:27:34.580
Groups are a set of elements
with a single operation, which
00:27:34.580 --> 00:27:35.830
we've been calling addition.
00:27:38.310 --> 00:27:42.500
A field is a set of elements
with two operations, which
00:27:42.500 --> 00:27:45.990
we'll call addition and
multiplication.
00:27:45.990 --> 00:27:46.990
So what do we have?
00:27:46.990 --> 00:27:50.720
We have a set of elements F.
We're going to be particularly
00:27:50.720 --> 00:27:53.030
interested where the
set is finite.
00:27:53.030 --> 00:27:55.272
Those are called
finite fields.
00:27:55.272 --> 00:28:00.190
And we're going to have two
operations, which I'll
00:28:00.190 --> 00:28:04.620
continue to write addition by
simple plus and multiplication
00:28:04.620 --> 00:28:07.740
with an asterisk, just to
be very explicit about
00:28:07.740 --> 00:28:08.695
everything.
00:28:08.695 --> 00:28:11.710
And after a while, you can
write these things as you
00:28:11.710 --> 00:28:15.170
would in ordinary arithmetic,
with just ordinary plus and
00:28:15.170 --> 00:28:18.550
juxtaposition for
multiplication.
00:28:18.550 --> 00:28:21.101
And what are the axioms
of a field?
00:28:21.101 --> 00:28:26.870
They're presented in an elegant
way in the notes,
00:28:26.870 --> 00:28:31.650
which obviously go back a long
way, but I got from Bob
00:28:31.650 --> 00:28:34.780
Gallager, and I like.
00:28:34.780 --> 00:28:36.120
All right.
00:28:36.120 --> 00:28:39.590
Under addition --
00:28:39.590 --> 00:28:42.710
so let's write it this way.
00:28:42.710 --> 00:28:47.500
Now, just considering the
addition operation is an
00:28:47.500 --> 00:28:50.720
abelian group.
00:28:50.720 --> 00:28:51.970
Commutative group.
00:28:57.370 --> 00:29:01.470
Which means it has an identity,
and we will continue
00:29:01.470 --> 00:29:05.620
to call that identity 0.
00:29:05.620 --> 00:29:09.970
Just as we do in the real
field, let's say.
00:29:09.970 --> 00:29:10.085
OK.
00:29:10.085 --> 00:29:12.540
Think of the real field, if you
like, as a model for all
00:29:12.540 --> 00:29:15.050
fields here.
00:29:15.050 --> 00:29:16.520
All right.
00:29:16.520 --> 00:29:19.130
So that's axiom one.
00:29:19.130 --> 00:29:22.250
Axiom two.
00:29:22.250 --> 00:29:26.320
If we take the non-zero elements
of the field, which I
00:29:26.320 --> 00:29:32.740
write by F star, explicitly
that's F not including 0 --
00:29:35.295 --> 00:29:38.550
not a very good notation,
but I'll use it --
00:29:38.550 --> 00:29:45.640
and the multiplication
operation, that, too is an
00:29:45.640 --> 00:29:46.890
abelian group.
00:29:52.590 --> 00:29:56.390
So this, of course, is why we
spent a little time on groups,
00:29:56.390 --> 00:30:00.850
abelian groups, so we'd
eventually be able to deal
00:30:00.850 --> 00:30:02.300
with fields.
00:30:02.300 --> 00:30:06.000
And its identity is called 1.
00:30:10.200 --> 00:30:12.300
Meaning that under
multiplication, 1 times
00:30:12.300 --> 00:30:14.445
anything is equal to itself.
00:30:18.030 --> 00:30:24.050
And then we have something
about how the operations
00:30:24.050 --> 00:30:35.450
distribute, the usual
distributive law that A times
00:30:35.450 --> 00:30:44.890
B plus C is equal to A times B
plus D times C, where I've
00:30:44.890 --> 00:30:46.670
written out all of these.
00:30:46.670 --> 00:30:51.290
So this is how addition and
multiplication interact again
00:30:51.290 --> 00:30:53.655
in a way that you're accustomed
to, and after a
00:30:53.655 --> 00:30:55.240
while, you don't need to write
all these parentheses.
00:30:58.770 --> 00:30:58.930
OK.
00:30:58.930 --> 00:31:03.090
So that's actually almost a
simpler set of axioms than for
00:31:03.090 --> 00:31:05.020
groups, once we understand
the group axioms.
00:31:07.830 --> 00:31:10.310
And so let's check.
00:31:10.310 --> 00:31:13.920
Is the real field, is the set
of all real numbers under
00:31:13.920 --> 00:31:15.500
ordinary real addition and
00:31:15.500 --> 00:31:19.955
multiplication, is that a field?
00:31:24.840 --> 00:31:26.040
What do we have to check?
00:31:26.040 --> 00:31:30.730
We have to check that under
addition, we're going to take
00:31:30.730 --> 00:31:33.240
the additive identity
as being equal to 0.
00:31:36.420 --> 00:31:41.800
Under our reduced set of group
axioms, the main thing we have
00:31:41.800 --> 00:31:46.060
to check is if we add any real
number to the reals, we get
00:31:46.060 --> 00:31:49.075
the reals again and the
one-to-one correspondence is
00:31:49.075 --> 00:31:50.355
the permutation.
00:31:50.355 --> 00:31:52.940
Is that correct?
00:31:52.940 --> 00:31:55.280
Yes, it is.
00:31:55.280 --> 00:31:59.840
And so this is OK.
00:31:59.840 --> 00:32:03.540
Now under multiplication, if
we take the non-zero real
00:32:03.540 --> 00:32:07.010
numbers, here's the question.
00:32:09.580 --> 00:32:17.240
If I have some alpha not equal
to 0, is alpha times the
00:32:17.240 --> 00:32:21.570
reals, not including 0 -- the
non-zero real numbers --
00:32:21.570 --> 00:32:23.250
equal to R star?
00:32:23.250 --> 00:32:27.180
And here I'm really implying a
one-to-one correspondence.
00:32:27.180 --> 00:32:29.670
So I might write it
more that way.
00:32:33.730 --> 00:32:39.150
I pose this question rather
abstractly, but you can easily
00:32:39.150 --> 00:32:40.790
convince yourself
that it's true.
00:32:44.270 --> 00:32:48.600
Any non-zero number, if I
multiply it by any non-zero
00:32:48.600 --> 00:32:51.270
number, I get a non-zero
number.
00:32:51.270 --> 00:32:53.480
Is the correspondence
one to one?
00:32:53.480 --> 00:32:59.820
Yes, because I can divide out
this number and get alpha.
00:33:02.810 --> 00:33:13.150
So alpha x on R star by
multiplication to give R star
00:33:13.150 --> 00:33:16.570
again, and this is a one-to-one
correspondence.
00:33:16.570 --> 00:33:19.350
But it's obvious why I have
to leave out 0, right?
00:33:19.350 --> 00:33:21.880
0 times any real number is 0.
00:33:21.880 --> 00:33:29.140
So at O, R star is simply
equal to zero set.
00:33:29.140 --> 00:33:32.360
So we always have to leave out
0 from multiplication.
00:33:32.360 --> 00:33:34.350
0 doesn't have an inverse.
00:33:34.350 --> 00:33:36.740
Everything else does
have an inverse.
00:33:36.740 --> 00:33:38.750
Under the standard group
operations, that's what we
00:33:38.750 --> 00:33:40.890
have to check.
00:33:40.890 --> 00:33:45.850
That would be the alternate
question, does every non-zero
00:33:45.850 --> 00:33:47.780
real number have an inverse?
00:33:47.780 --> 00:33:49.675
That's easier to see,
the answer is yes.
00:33:49.675 --> 00:33:51.810
Multiplicative inverse.
00:33:51.810 --> 00:33:55.568
Inverse of alpha is
1 over alpha.
00:33:55.568 --> 00:33:57.903
AUDIENCE: [INAUDIBLE]
00:33:57.903 --> 00:33:58.530
PROFESSOR: Yes.
00:33:58.530 --> 00:34:02.050
I've used the alternative set
of axioms, including the
00:34:02.050 --> 00:34:03.300
permutation property.
00:34:08.940 --> 00:34:11.020
To check whether there's an
abelian group, I've asked if
00:34:11.020 --> 00:34:13.929
alpha R star is the permutation
of R star.
00:34:13.929 --> 00:34:18.770
And without going through
details, I claim it is.
00:34:18.770 --> 00:34:20.020
Thank you.
00:34:22.620 --> 00:34:23.170
All right.
00:34:23.170 --> 00:34:25.159
So we checked that.
00:34:25.159 --> 00:34:27.750
Of course, the distributive
law holds.
00:34:27.750 --> 00:34:33.670
So the real field is a field,
which you probably were
00:34:33.670 --> 00:34:35.115
willing to accept on
faith, anyway.
00:34:37.909 --> 00:34:40.500
Similarly, you go through
exactly the same arguments for
00:34:40.500 --> 00:34:41.750
the complex field.
00:34:45.330 --> 00:34:48.060
What about the binary field?
00:34:48.060 --> 00:34:51.449
We think we understand
that by now.
00:34:51.449 --> 00:34:59.740
Here the operations are mod-2
addition and mod-2
00:34:59.740 --> 00:35:00.905
multiplication.
00:35:00.905 --> 00:35:04.600
I've written down explicitly
the addition and
00:35:04.600 --> 00:35:05.850
multiplication tables.
00:35:11.410 --> 00:35:16.701
Under addition, we simply
have Z2 again.
00:35:19.347 --> 00:35:20.220
F2.
00:35:20.220 --> 00:35:23.446
The additive group of
F2 is simply Z2.
00:35:23.446 --> 00:35:27.030
We forget about multiplication.
00:35:27.030 --> 00:35:31.220
We've seen quite a few times
now that that's a group.
00:35:31.220 --> 00:35:35.580
Under multiplication, what are
the non-zero elements of F2?
00:35:39.910 --> 00:35:41.160
Just one element. one.
00:35:43.870 --> 00:35:46.230
This includes the identity?
00:35:46.230 --> 00:35:48.250
Yes.
00:35:48.250 --> 00:35:50.245
Is it a group?
00:35:50.245 --> 00:35:50.660
Yeah.
00:35:50.660 --> 00:35:51.750
It's a trivial group.
00:35:51.750 --> 00:35:53.160
1 times 1 equals 1.
00:35:56.620 --> 00:36:00.060
1 under multiplication is
isomorphic to the trivial
00:36:00.060 --> 00:36:02.630
group 0 under addition.
00:36:02.630 --> 00:36:07.720
Its group table is
1 times 1 is 1.
00:36:11.750 --> 00:36:13.110
Sure enough.
00:36:13.110 --> 00:36:15.560
That's the identity
permutation.
00:36:15.560 --> 00:36:17.482
Sometimes when things get
too trivial, it's a
00:36:17.482 --> 00:36:18.170
little hard to check.
00:36:18.170 --> 00:36:20.760
But yes.
00:36:20.760 --> 00:36:25.180
And distributive is
easy to check.
00:36:25.180 --> 00:36:26.430
OK?
00:36:28.980 --> 00:36:36.840
So that's all it takes
to define a field.
00:36:36.840 --> 00:36:40.600
Of course, by the inverse
property, when we have
00:36:40.600 --> 00:36:44.940
addition, this also implies an
inverse and a subtraction
00:36:44.940 --> 00:36:49.260
operation and a cancellation
and additive identities.
00:36:49.260 --> 00:36:55.350
We have a field element on both
sides of a plus b equals
00:36:55.350 --> 00:36:58.620
a plus c, then b plus
b equals c.
00:36:58.620 --> 00:37:00.890
That's what I mean
by cancellation.
00:37:00.890 --> 00:37:04.010
Similarly under multiplication,
we get a
00:37:04.010 --> 00:37:05.950
multiplicative inverse.
00:37:05.950 --> 00:37:11.320
1 over alpha, for any alpha in
F. We, therefore, are able to
00:37:11.320 --> 00:37:14.490
define division.
00:37:14.490 --> 00:37:21.880
And we have cancellation for
multiplicative identity.
00:37:21.880 --> 00:37:25.580
So we immediately get a lot from
these group properties.
00:37:25.580 --> 00:37:27.765
We get all the properties
you expect of fields.
00:37:27.765 --> 00:37:30.700
You can add, subtract, multiply,
or divide, all in
00:37:30.700 --> 00:37:32.670
the usual way that we do
over the real field.
00:37:36.900 --> 00:37:38.150
OK.
00:37:40.192 --> 00:37:41.530
Let's stay over here.
00:37:48.480 --> 00:37:51.736
I think my next topic
is prime fields.
00:37:59.990 --> 00:38:00.190
Yes.
00:38:00.190 --> 00:38:01.880
So prime fields.
00:38:01.880 --> 00:38:04.490
When we talked about the
factorization properties of
00:38:04.490 --> 00:38:08.630
the integers, we talked
about primes p.
00:38:08.630 --> 00:38:17.610
And now I'm going to talk about
Fp is going to be a
00:38:17.610 --> 00:38:20.990
field with a finite number
of elements where
00:38:20.990 --> 00:38:24.190
the number is a prime.
00:38:24.190 --> 00:38:27.140
So what are the elements in
this field going to be?
00:38:27.140 --> 00:38:34.890
Are they simply going to be 0,
1 up through p minus 1 again,
00:38:34.890 --> 00:38:40.240
the same elements as were in
the cyclic group with p
00:38:40.240 --> 00:38:44.180
elements where I'm restricting
m now to be a prime p?
00:38:47.100 --> 00:38:54.120
And for my addition operation
and my multiplication
00:38:54.120 --> 00:38:58.650
operation, I'm going to just
let these be mod-p addition
00:38:58.650 --> 00:39:00.613
and multiplication now.
00:39:05.930 --> 00:39:08.680
And I claim that this
is a field.
00:39:14.810 --> 00:39:19.440
So actually, the proof follows
very close to what I
00:39:19.440 --> 00:39:21.620
just did for F2.
00:39:21.620 --> 00:39:22.905
F2 is a model for this.
00:39:25.840 --> 00:39:28.720
But it's a little harder
to check this.
00:39:28.720 --> 00:39:35.710
Under a, under the addition
operation, Fp really is just
00:39:35.710 --> 00:39:39.360
Zp again, so that's OK.
00:39:39.360 --> 00:39:41.840
That's an abelian group.
00:39:41.840 --> 00:39:43.090
Zmod-p.
00:39:45.010 --> 00:39:47.120
Or the quotient group,
Zmod-pZ.
00:39:49.790 --> 00:39:58.110
We can also again think of this
as Zmod-pZ, if we want.
00:40:01.720 --> 00:40:05.080
So everything is going to become
mod-Z. That's a very
00:40:05.080 --> 00:40:06.720
useful way of thinking of it.
00:40:06.720 --> 00:40:09.810
So we're really thinking of
these as remainders or
00:40:09.810 --> 00:40:18.240
representatives of the residue
classes of pZ in Z. This is
00:40:18.240 --> 00:40:24.610
pZ, this is pZ plus 1 up
to Z minus 1, up to pZ
00:40:24.610 --> 00:40:26.450
plus p minus 1.
00:40:26.450 --> 00:40:29.790
This is the same
as pZ minus 1.
00:40:29.790 --> 00:40:30.190
OK.
00:40:30.190 --> 00:40:34.920
The real question is, if we take
the non-zero elements of
00:40:34.920 --> 00:40:42.915
Fp, is this closed?
00:40:46.110 --> 00:40:50.860
And does every element
have an inverse?
00:40:50.860 --> 00:40:54.130
Or equivalently, when we
multiply by a particular
00:40:54.130 --> 00:40:56.400
element, do we just get
a permutation of this?
00:41:00.250 --> 00:41:02.280
The reason that p has
to be a prime --
00:41:02.280 --> 00:41:08.040
let's suppose we take two of
these things, a and b, and we
00:41:08.040 --> 00:41:09.290
multiply them.
00:41:11.820 --> 00:41:13.305
What's the multiplicative
rule?
00:41:21.800 --> 00:41:26.095
a times b is just ab mod-p.
00:41:26.095 --> 00:41:29.380
That's what I defined
multiplication as.
00:41:29.380 --> 00:41:33.660
Now the question is, could
that possibly be 0?
00:41:33.660 --> 00:41:38.530
Which is the same as saying,
could a times b be a multiple
00:41:38.530 --> 00:41:43.760
of p, where a and b are taken
from the non-zero
00:41:43.760 --> 00:41:46.460
elements of the field?
00:41:46.460 --> 00:41:49.970
And here, because p is a prime,
it's clear that you
00:41:49.970 --> 00:41:55.410
can't multiply two non-zero
numbers which are less than p
00:41:55.410 --> 00:41:57.870
and get a multiple
of the prime p.
00:42:02.120 --> 00:42:08.190
If p were not a prime,
then you could.
00:42:08.190 --> 00:42:12.570
If we took n equals 10 again,
let's say, and we multiplied 2
00:42:12.570 --> 00:42:18.390
times 5 from the 10 elements
of these residue classes, 2
00:42:18.390 --> 00:42:25.370
times 5 is, in fact, equal to
0 mod-10, and therefore Fp
00:42:25.370 --> 00:42:30.040
star, or F10 star, would
not be closed under
00:42:30.040 --> 00:42:30.930
multiplication.
00:42:30.930 --> 00:42:32.140
We would get a 0.
00:42:32.140 --> 00:42:35.550
But in this case, we easily
prove, because it's a prime,
00:42:35.550 --> 00:42:37.930
that it's not equal to 0.
00:42:37.930 --> 00:42:42.930
And therefore it's in Fp star,
so it is closed under
00:42:42.930 --> 00:42:44.180
multiplication.
00:42:46.560 --> 00:42:50.050
And the other thing we have to
check is that it's one-to-one.
00:42:50.050 --> 00:43:02.080
In other words, can a star b
equal to a star c, and by the
00:43:02.080 --> 00:43:05.830
cancellation property,
which holds in --
00:43:09.620 --> 00:43:09.930
Sorry.
00:43:09.930 --> 00:43:13.450
We've got to establish the
cancellation property holds
00:43:13.450 --> 00:43:18.140
under mod-b arithmetic, but
it does, and so we get the
00:43:18.140 --> 00:43:20.900
cancellation property, that
this is true if and
00:43:20.900 --> 00:43:23.340
only if b equals c.
00:43:23.340 --> 00:43:28.530
So in other words, as we run
through all of these multiples
00:43:28.530 --> 00:43:31.910
for any particular alpha,
we're going to get a
00:43:31.910 --> 00:43:32.690
permutation.
00:43:32.690 --> 00:43:34.290
We need to get the
same set back.
00:43:34.290 --> 00:43:35.820
Everything is finite.
00:43:35.820 --> 00:43:38.960
We're going to get a bunch of
distinct elements of the same
00:43:38.960 --> 00:43:41.310
size as the set itself.
00:43:41.310 --> 00:43:45.920
Therefore, it has to
be the set again.
00:43:45.920 --> 00:43:47.910
I haven't said that
very well again.
00:43:47.910 --> 00:43:48.980
That's why we have notes.
00:43:48.980 --> 00:43:53.090
It's written up correctly
in the notes.
00:43:53.090 --> 00:43:56.390
But we have basically checked
everything that we need to
00:43:56.390 --> 00:44:01.580
check, showed that Fp star
is an abelian group under
00:44:01.580 --> 00:44:06.690
multiplication when p is a
prime, and clearly not when p
00:44:06.690 --> 00:44:08.085
is not a prime.
00:44:08.085 --> 00:44:10.065
AUDIENCE: [INAUDIBLE]
00:44:10.065 --> 00:44:13.400
the inverse, there
is inverse of a?
00:44:13.400 --> 00:44:14.650
PROFESSOR: Yes.
00:44:16.610 --> 00:44:19.950
But basically, we have to prove
that if I take any of
00:44:19.950 --> 00:44:25.790
these, if I take a particular
one, say, alpha, and multiply
00:44:25.790 --> 00:44:31.990
times all of them in Fp star,
that I'm just going to get Fp
00:44:31.990 --> 00:44:34.080
star again.
00:44:34.080 --> 00:44:38.940
And to prove that, I have to
prove that alpha times a is
00:44:38.940 --> 00:44:47.710
not equal to alpha times
b if a not equal to b.
00:44:47.710 --> 00:44:49.700
That's all I need
to prove, right?
00:44:49.700 --> 00:44:52.370
And that comes from the
properties of mod-p
00:44:52.370 --> 00:44:53.000
arithmetic.
00:44:53.000 --> 00:44:54.865
That is what is to be proved.
00:44:54.865 --> 00:44:58.232
I need to use mod-p arithmetic
to prove that.
00:44:58.232 --> 00:44:59.482
AUDIENCE: [INAUDIBLE]
00:45:03.080 --> 00:45:03.650
PROFESSOR: Oh.
00:45:03.650 --> 00:45:08.050
I have to check the identity
is in here.
00:45:08.050 --> 00:45:09.630
The identity is in here.
00:45:09.630 --> 00:45:10.230
It has one.
00:45:10.230 --> 00:45:10.670
I'm sorry.
00:45:10.670 --> 00:45:12.175
I should have checked
that, too.
00:45:12.175 --> 00:45:15.060
But 1 is the identity
for multiplication.
00:45:15.060 --> 00:45:19.380
And then from this property,
since we multiply alpha p
00:45:19.380 --> 00:45:22.390
star, we get Fp star again.
00:45:22.390 --> 00:45:25.080
That includes one.
00:45:25.080 --> 00:45:25.490
All right?
00:45:25.490 --> 00:45:29.940
So it's got to be one of these
guys which, times alpha, gives
00:45:29.940 --> 00:45:33.200
1, and that shows the existence
of an inverse.
00:45:33.200 --> 00:45:38.080
So you can do it any
way you want.
00:45:38.080 --> 00:45:42.130
But the key to the proof is to
prove this, and that's why I
00:45:42.130 --> 00:45:45.720
focused on the permutation
property.
00:45:45.720 --> 00:45:47.850
Permutation property is
really what you prove
00:45:47.850 --> 00:45:49.100
to demonstrate this.
00:45:53.390 --> 00:45:54.150
OK?
00:45:54.150 --> 00:45:55.900
Good.
00:45:55.900 --> 00:45:58.670
Everyone seems to be following
closely here.
00:45:58.670 --> 00:46:00.335
Any further questions?
00:46:00.335 --> 00:46:03.020
This is important, because
we've got our
00:46:03.020 --> 00:46:05.360
first finite field.
00:46:05.360 --> 00:46:09.640
The integers mod-p are a finite
field of size p for any
00:46:09.640 --> 00:46:11.110
prime state.
00:46:11.110 --> 00:46:15.540
We've got F2, F3, F5,
F7, and so forth.
00:46:19.830 --> 00:46:22.370
OK.
00:46:22.370 --> 00:46:23.855
Further on this subject.
00:46:31.620 --> 00:46:34.670
We have two closely related
propositions.
00:46:34.670 --> 00:46:57.100
One, every finite field with
prime p elements is
00:46:57.100 --> 00:46:58.840
isomorphic to Fp.
00:47:02.230 --> 00:47:05.690
So if you give me a finite
field, you tell me it has p
00:47:05.690 --> 00:47:09.980
elements, I'll show you that
it basically has the same
00:47:09.980 --> 00:47:11.940
addition and multiplication
tables with relabeling.
00:47:16.460 --> 00:47:29.060
And secondly, every finite field
with an arbitrary number
00:47:29.060 --> 00:47:38.970
of elements, for every finite
field, the integers of the
00:47:38.970 --> 00:47:52.330
field form a prime field for
some P. You understand my
00:47:52.330 --> 00:47:55.250
abbreviations.
00:47:55.250 --> 00:47:58.010
And the proofs of these are
very closely related.
00:47:58.010 --> 00:48:05.290
What do I mean by the integers
of a field, of a finite field?
00:48:13.530 --> 00:48:14.150
OK.
00:48:14.150 --> 00:48:20.270
Well, let's start from the
very most basic thing.
00:48:20.270 --> 00:48:21.590
What do we know?
00:48:21.590 --> 00:48:24.730
We know that the field contains
0 and 1, and those
00:48:24.730 --> 00:48:27.760
are going to be two of the
integers of the field.
00:48:27.760 --> 00:48:32.120
So 0 and 1 are in F.
00:48:32.120 --> 00:48:39.060
Let's use the closure
under addition.
00:48:39.060 --> 00:48:44.995
Clearly 1 plus 1 is in F. We're
going to call that 2.
00:48:47.630 --> 00:48:54.420
1 plus 1 plus 1 is in F. We're
going to call that 3.
00:48:54.420 --> 00:48:55.670
And so forth.
00:48:59.620 --> 00:49:04.480
And of course, since the field
is finite, eventually this is
00:49:04.480 --> 00:49:07.240
going to have to repeat.
00:49:07.240 --> 00:49:16.700
And from the fact it repeats,
you're basically going to show
00:49:16.700 --> 00:49:19.920
that at some point, one
of these is going
00:49:19.920 --> 00:49:22.000
to be equal to 0.
00:49:22.000 --> 00:49:25.560
So there's going to be some n.
00:49:25.560 --> 00:49:27.690
The first repeat is
going to be n
00:49:27.690 --> 00:49:38.430
equal to 0 in F. OK.
00:49:38.430 --> 00:49:39.870
So that's what I mean
by the integers.
00:49:45.490 --> 00:49:59.260
The integers clearly form a
subgroup of the additive group
00:49:59.260 --> 00:50:02.940
of F, to form a subgroup
under addition.
00:50:09.540 --> 00:50:12.806
And in fact, a cyclic
subgroup.
00:50:12.806 --> 00:50:18.800
I'm skipping over some of the
details here, but that's a
00:50:18.800 --> 00:50:23.240
claim at this point that I
haven't really demonstrated.
00:50:23.240 --> 00:50:27.890
But just from a subgroup
property, let's attack number
00:50:27.890 --> 00:50:30.040
one up here.
00:50:30.040 --> 00:50:35.030
Suppose we have a field with p
elements, and the additive
00:50:35.030 --> 00:50:38.310
group of the field
has p elements.
00:50:38.310 --> 00:50:42.110
It consists of the
same elements.
00:50:42.110 --> 00:50:46.000
And by Lagrange's theorem, what
are the possible orders
00:50:46.000 --> 00:50:47.485
of that subgroup?
00:50:50.090 --> 00:50:52.780
What are the possible number of
elements in that subgroup,
00:50:52.780 --> 00:50:54.980
the sizes of the subgroup?
00:50:54.980 --> 00:50:56.315
The order of a group
is its size.
00:50:59.670 --> 00:51:01.460
Well, it has to divide p.
00:51:01.460 --> 00:51:04.095
There aren't many things
that divide a prime p.
00:51:04.095 --> 00:51:06.710
There's 1 and there's p.
00:51:06.710 --> 00:51:07.140
OK?
00:51:07.140 --> 00:51:12.960
So the subgroup either has
a single element or
00:51:12.960 --> 00:51:15.530
it's all of the group.
00:51:18.040 --> 00:51:21.020
If there's a single element
-- let's to keep an
00:51:21.020 --> 00:51:24.520
open mind here --
00:51:24.520 --> 00:51:31.350
then what that means is that
if I take G and I add it to
00:51:31.350 --> 00:51:33.750
itself, since it's a subgroup,
it has to give an
00:51:33.750 --> 00:51:35.360
element of the group.
00:51:35.360 --> 00:51:39.140
But there is only one element
of the group.
00:51:39.140 --> 00:51:43.590
Let's say G is the single
element in this subgroup.
00:51:43.590 --> 00:51:45.310
I guess it could only be 1.
00:51:45.310 --> 00:51:46.750
Let's start out with one.
00:51:46.750 --> 00:51:50.480
So suppose one is the only
element of the subgroup.
00:51:50.480 --> 00:51:55.560
Then I get the equation 1 plus
1 equals 1, which by
00:51:55.560 --> 00:52:01.090
cancellation implies
that 1 equals 0.
00:52:01.090 --> 00:52:03.250
OK, well, that can't be true.
00:52:03.250 --> 00:52:06.490
In a field the multiplicative
identity is not
00:52:06.490 --> 00:52:08.550
the additive identity.
00:52:08.550 --> 00:52:13.610
So that can't be true.
00:52:13.610 --> 00:52:17.370
That would only be true if we
had a field with one element,
00:52:17.370 --> 00:52:21.340
and fields implicitly always
have at least two
00:52:21.340 --> 00:52:23.470
elements, 0 and 1.
00:52:23.470 --> 00:52:26.300
F2 is the smallest
finite field.
00:52:26.300 --> 00:52:29.390
I suppose we could set up a
single element that sort of
00:52:29.390 --> 00:52:32.590
satisfies all these axioms,
but then, what is the
00:52:32.590 --> 00:52:34.100
multiplicative group?
00:52:34.100 --> 00:52:35.460
All right.
00:52:35.460 --> 00:52:37.660
So this can't happen.
00:52:37.660 --> 00:52:41.710
So that means this subgroup
has to have p elements.
00:52:41.710 --> 00:52:44.370
It has to consist of all the
elements of the field.
00:52:44.370 --> 00:52:47.310
So that means the integers are
all the elements of the field.
00:52:50.600 --> 00:52:58.220
But now the isomorphism, then,
is that this is isomorphic Fp
00:52:58.220 --> 00:52:59.470
under the isomorphism.
00:53:01.560 --> 00:53:05.500
This corresponds to 2, this
corresponds to 3, and so
00:53:05.500 --> 00:53:06.980
forth, in Fp.
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,
00:53:12.190 --> 00:53:16.170
so 2 plus 3 is going
to be 5 1's.
00:53:18.690 --> 00:53:22.410
Mod size the field, whenever
this cycles.
00:53:22.410 --> 00:53:25.510
So this is going to
have to be p, and
00:53:25.510 --> 00:53:27.570
basically, that shows --
00:53:27.570 --> 00:53:28.820
AUDIENCE: [INAUDIBLE]
00:53:30.670 --> 00:53:33.288
to prove that it is isomorphical
[UNINTELLIGIBLE]
00:53:33.288 --> 00:53:37.536
multiplying 1 plus 1 into
1 plus 1 plus 1.
00:53:37.536 --> 00:53:38.810
But it is typical --
00:53:38.810 --> 00:53:40.240
PROFESSOR: I really
have only used the
00:53:40.240 --> 00:53:41.380
additive property here.
00:53:41.380 --> 00:53:43.884
I don't think multiplication
enters into it.
00:53:47.300 --> 00:53:52.500
OK, here's where the
multiplicative
00:53:52.500 --> 00:53:53.610
property adds in.
00:53:53.610 --> 00:53:59.320
I have to prove not only that
this is isomorphic to Fp as an
00:53:59.320 --> 00:54:11.160
additive group, but the
multiplication tables are
00:54:11.160 --> 00:54:14.690
isomorphic under the
same relabeling.
00:54:14.690 --> 00:54:19.410
So for that, I have to show
that 2 times 3, when I've
00:54:19.410 --> 00:54:22.250
defined 3 and 3 this way, gives
me the same result as
00:54:22.250 --> 00:54:26.500
multiplying 2 and
3 in F p mod-p.
00:54:26.500 --> 00:54:31.530
But again, I could do this just
because sort of mod-p
00:54:31.530 --> 00:54:35.840
commutes with addition
and multiplication.
00:54:35.840 --> 00:54:42.960
If I multiply 1 plus 1, two 1's
times three 1's, so I'm
00:54:42.960 --> 00:54:47.630
going to get six 1's, and that's
exactly what I get in
00:54:47.630 --> 00:54:51.290
Fp, reducing everything mod-p.
00:54:51.290 --> 00:54:57.500
So I have to check that also
to prove this isomorphism.
00:54:57.500 --> 00:55:00.070
And this is done carefully
in the notes.
00:55:00.070 --> 00:55:02.850
The distributive law holds
because the distributive law
00:55:02.850 --> 00:55:07.090
holds for sums of n 1's.
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
00:55:14.680 --> 00:55:18.255
same regardless of where you
put it in, how you put the
00:55:18.255 --> 00:55:20.150
parentheses.
00:55:20.150 --> 00:55:20.570
OK.
00:55:20.570 --> 00:55:27.230
So with some sorry hand-waving
here, we've basically given
00:55:27.230 --> 00:55:29.750
the idea of how to prove this.
00:55:29.750 --> 00:55:33.890
It's basically Lagrange that the
additive subgroup has to
00:55:33.890 --> 00:55:42.360
be of size 1 or p, and we prove
quickly that actually p
00:55:42.360 --> 00:55:44.360
is the only case that works.
00:55:44.360 --> 00:55:48.260
And then we extend all the
arithmetic properties by just
00:55:48.260 --> 00:55:51.888
observing they'll hold
for 1 plus 1 plus 1.
00:55:51.888 --> 00:55:52.382
Yeah?
00:55:52.382 --> 00:55:54.852
AUDIENCE: [INAUDIBLE]
00:55:54.852 --> 00:55:58.580
the line 1 plus 1 plus
1 like that?
00:55:58.580 --> 00:55:59.830
Might we [UNINTELLIGIBLE PHRASE]
00:56:03.794 --> 00:56:05.700
1 is equal to 0.
00:56:05.700 --> 00:56:08.270
What is actually
[UNINTELLIGIBLE PHRASE]?
00:56:08.270 --> 00:56:11.135
Should we state that 1 has
to be different than 0?
00:56:11.135 --> 00:56:11.560
PROFESSOR: Yeah.
00:56:11.560 --> 00:56:15.360
I guess I could simply get
around that by stating that
00:56:15.360 --> 00:56:19.270
the multiplicative identity has
to be different from the
00:56:19.270 --> 00:56:20.260
additive identity.
00:56:20.260 --> 00:56:23.710
It clearly follows from this,
and I think I put it as an
00:56:23.710 --> 00:56:28.930
exercise, 0 times any group
element has got
00:56:28.930 --> 00:56:31.600
to be equal to 0.
00:56:31.600 --> 00:56:33.360
So this is how 0 behaves under
00:56:33.360 --> 00:56:37.150
multiplication from these axioms.
00:56:37.150 --> 00:56:40.520
But 1, as the multiplicative
identity, has to satisfy that
00:56:40.520 --> 00:56:45.960
rule, so clearly, 0
cannot equal to 1.
00:56:45.960 --> 00:56:49.250
Unless, in some trivial sense,
there is only one element in
00:56:49.250 --> 00:56:52.580
the groups if there's any
non-zero element.
00:56:52.580 --> 00:56:56.860
So this implies that 0
is not equal to 1.
00:56:56.860 --> 00:56:59.092
Just could have included
that as an axiom.
00:57:02.068 --> 00:57:02.564
Yeah?
00:57:02.564 --> 00:57:07.028
AUDIENCE: Assume a
and b [INAUDIBLE]
00:57:07.028 --> 00:57:08.020
PROFESSOR: Excuse me?
00:57:08.020 --> 00:57:11.530
AUDIENCE: Assume a and
b, [UNINTELLIGIBLE]
00:57:11.530 --> 00:57:13.620
does not include 0?
00:57:13.620 --> 00:57:14.870
PROFESSOR: Correct.
00:57:17.060 --> 00:57:17.810
Yeah, you're right.
00:57:17.810 --> 00:57:18.440
OK.
00:57:18.440 --> 00:57:25.155
So it follows from this
that 1 is not 0.
00:57:25.155 --> 00:57:26.100
Yeah.
00:57:26.100 --> 00:57:28.740
I'm sorry I don't personally
have a lot of patience for
00:57:28.740 --> 00:57:30.810
these fine details.
00:57:30.810 --> 00:57:33.430
For mathematicians,
it's important to
00:57:33.430 --> 00:57:38.340
keep them all in mind.
00:57:38.340 --> 00:57:41.830
But my effort is to make these
propositions plausible enough
00:57:41.830 --> 00:57:45.800
so that you can believe them,
and you can go back and read a
00:57:45.800 --> 00:57:50.410
real proof and see that the
proof must be correct
00:57:50.410 --> 00:57:53.360
intuitively, without just
mechanically checking it.
00:58:00.070 --> 00:58:02.410
OK.
00:58:02.410 --> 00:58:07.470
Let me just again outline
how this works.
00:58:07.470 --> 00:58:09.260
It's very similar.
00:58:09.260 --> 00:58:12.800
Again, given any finite field,
if we define the integers of
00:58:12.800 --> 00:58:18.870
the field in this way, we show
that eventually they form a
00:58:18.870 --> 00:58:19.930
cyclic group.
00:58:19.930 --> 00:58:22.735
Their cyclic group
is something that
00:58:22.735 --> 00:58:23.820
has a single generator.
00:58:23.820 --> 00:58:25.630
The generator is 1.
00:58:25.630 --> 00:58:30.880
So eventually it has to cycle
for some number n.
00:58:30.880 --> 00:58:35.740
Now could n be a non-prime?
00:58:35.740 --> 00:58:41.110
No, because this is a field, and
if n were non-prime, then
00:58:41.110 --> 00:58:44.040
we would be able to find two
integers that multiplied
00:58:44.040 --> 00:58:47.960
together gave 0.
00:58:47.960 --> 00:58:53.600
And that's forbidden by the
axioms of the field.
00:58:53.600 --> 00:58:59.970
So the only possibility is that
n is a prime, and in that
00:58:59.970 --> 00:59:05.040
case, we have found what's
called a subfield, a subset of
00:59:05.040 --> 00:59:07.960
the elements of the field which
itself is a field under
00:59:07.960 --> 00:59:09.210
the field axioms.
00:59:11.340 --> 00:59:14.050
And the field has p elements,
and we already know that every
00:59:14.050 --> 00:59:19.150
finite field with p elements
is isomorphic to Fp.
00:59:19.150 --> 00:59:25.220
So it can only be that the set
of integers is a subfield
00:59:25.220 --> 00:59:28.030
which is isomorphic to
Fp for some prime p.
00:59:30.670 --> 00:59:31.990
OK?
00:59:31.990 --> 00:59:36.320
So within any finite field,
we're always going to find,
00:59:36.320 --> 00:59:41.090
just by writing out the integers
and seeing how they
00:59:41.090 --> 00:59:44.140
behave under the additive
property that there are
00:59:44.140 --> 00:59:45.390
exactly p of them.
00:59:51.280 --> 00:59:54.490
So every finite field has
a prime called the
00:59:54.490 --> 00:59:55.276
characteristic.
00:59:55.276 --> 00:59:57.370
The prime characteristic
of the field.
00:59:57.370 --> 00:59:59.760
This is defined as the
characteristic.
00:59:59.760 --> 01:00:03.580
The size of the integer subfield
is the characteristic
01:00:03.580 --> 01:00:05.410
of the field.
01:00:05.410 --> 01:00:11.030
And it has an interesting
property, a
01:00:11.030 --> 01:00:12.280
very important property.
01:00:12.280 --> 01:00:17.480
Suppose we take this p and we
multiply it by any field
01:00:17.480 --> 01:00:21.190
element called data
in the field.
01:00:25.310 --> 01:00:28.420
By the distributive law,
this is just equal to
01:00:28.420 --> 01:00:32.110
1 plus 1 plus --
01:00:32.110 --> 01:00:33.780
so what do I mean by this?
01:00:33.780 --> 01:00:37.410
I mean beta --
01:00:37.410 --> 01:00:40.980
whenever I write an integer
times a field element, I mean
01:00:40.980 --> 01:00:46.220
beta plus beta plus
so forth, p times.
01:00:49.880 --> 01:00:56.430
But this is equal to 1 plus 1,
so forth, by the distributive
01:00:56.430 --> 01:01:03.670
law, I guess, times
theta, p times.
01:01:03.670 --> 01:01:04.920
And what is this equal to?
01:01:08.260 --> 01:01:09.290
This is equal to 0.
01:01:09.290 --> 01:01:13.200
So this is equal to 0 times
beta, which fortunately I just
01:01:13.200 --> 01:01:15.585
told you always must equal 0.
01:01:21.360 --> 01:01:21.880
OK.
01:01:21.880 --> 01:01:28.840
So the conclusion is that if
we add any field element to
01:01:28.840 --> 01:01:35.860
itself p times, we're going to
get 0 for all beta in the
01:01:35.860 --> 01:01:39.870
field where p is the
characteristic of the field.
01:01:43.700 --> 01:01:49.560
Now in digital communications,
we're almost always dealing
01:01:49.560 --> 01:01:52.690
with a case where the
characteristic of the field is
01:01:52.690 --> 01:01:53.760
going to be 2.
01:01:53.760 --> 01:01:55.940
The prime subfield is just
going to be the two
01:01:55.940 --> 01:01:57.620
elements 0 and 1.
01:01:57.620 --> 01:02:01.290
1 plus 1 is going to
be equal to 0.
01:02:01.290 --> 01:02:04.450
So subtraction will be
the same as addition.
01:02:04.450 --> 01:02:11.420
And in that particular case, we
will have that the sum beta
01:02:11.420 --> 01:02:14.950
plus beta of any 2 field
elements in a field of
01:02:14.950 --> 01:02:18.930
characteristic two is going
to be equal to 0.
01:02:18.930 --> 01:02:21.910
Just as we had for code words
in binary linear codes.
01:02:25.470 --> 01:02:28.980
Binary linear codes are not
fields, they're vector spaces,
01:02:28.980 --> 01:02:31.160
but it's a similar
property here.
01:02:31.160 --> 01:02:35.900
You add any element of field of
characteristic 2 to itself,
01:02:35.900 --> 01:02:37.300
and you're going to get 0.
01:02:37.300 --> 01:02:41.450
So this shows that addition is
the same as subtraction.
01:02:41.450 --> 01:02:47.290
Beta equals minus beta in a
field of characteristic 0.
01:02:47.290 --> 01:02:50.370
Which is a little bit
more general.
01:02:53.930 --> 01:02:54.025
OK.
01:02:54.025 --> 01:03:00.170
So we have some fields now, and
we find these fields are
01:03:00.170 --> 01:03:05.750
the only field of prime size,
and that every finite field
01:03:05.750 --> 01:03:10.700
has an important subfield
and a prime subfield.
01:03:10.700 --> 01:03:17.290
And that has important
properties, consequences for
01:03:17.290 --> 01:03:19.870
the field itself.
01:03:19.870 --> 01:03:20.550
All right.
01:03:20.550 --> 01:03:22.290
I think that's everything
I want to
01:03:22.290 --> 01:03:27.170
say about prime fields.
01:03:27.170 --> 01:03:31.560
Now we go on to the next
important algebraic object,
01:03:31.560 --> 01:03:32.810
polynomials.
01:03:37.690 --> 01:03:45.160
And again, it's hard to know
just how detailed to be,
01:03:45.160 --> 01:03:51.010
because of course you've all
seen polynomials, and you
01:03:51.010 --> 01:03:54.520
intuitively or formally know
something about their
01:03:54.520 --> 01:03:56.830
algebraic properties,
their factorization
01:03:56.830 --> 01:03:58.630
properties, and so forth.
01:03:58.630 --> 01:04:02.200
So I'm going to go pretty
quickly, and this will be in
01:04:02.200 --> 01:04:03.450
the nature of a review.
01:04:07.350 --> 01:04:08.600
A polynomial --
01:04:12.130 --> 01:04:14.020
maybe the simplest way --
01:04:14.020 --> 01:04:16.650
how do you define
a polynomial?
01:04:16.650 --> 01:04:18.670
What does it look like?
01:04:18.670 --> 01:04:20.760
It looks like this.
01:04:20.760 --> 01:04:29.900
F0 plus F1 times x plus F2 times
x squared, so forth,
01:04:29.900 --> 01:04:33.120
plus Fm times x to the m.
01:04:33.120 --> 01:04:36.800
That's what it looks like if
it's a non-zero polynomial.
01:04:40.010 --> 01:04:40.900
Or even if it's just --
01:04:40.900 --> 01:04:44.800
you could consider all 0
coefficients to be the zero
01:04:44.800 --> 01:04:45.420
polynomial.
01:04:45.420 --> 01:04:51.560
But in general, the convention
is, we write f of x equal to
01:04:51.560 --> 01:04:55.070
that if x is non-zero.
01:04:55.070 --> 01:04:56.730
What are these f's?
01:04:56.730 --> 01:04:58.510
These are called the
coefficients of the
01:04:58.510 --> 01:05:00.050
polynomial.
01:05:00.050 --> 01:05:03.510
And where do they live?
01:05:03.510 --> 01:05:08.160
We need the coefficients to
be in some common field.
01:05:11.680 --> 01:05:15.010
You've often seen these in the
real or complex field.
01:05:15.010 --> 01:05:17.610
Here they're going to
be in finite fields.
01:05:17.610 --> 01:05:19.310
In particular, very
shortly, they're
01:05:19.310 --> 01:05:22.180
going to be prime fields.
01:05:22.180 --> 01:05:26.480
But in general, we'll just
say these f's have
01:05:26.480 --> 01:05:29.290
to be in some field.
01:05:29.290 --> 01:05:37.460
So we're talking about a
polynomial over F where F is
01:05:37.460 --> 01:05:38.590
some field.
01:05:38.590 --> 01:05:40.640
So there's always some
underlying field if there
01:05:40.640 --> 01:05:42.730
isn't a vector space.
01:05:42.730 --> 01:05:46.970
Some similarities between
this and vector spaces.
01:05:46.970 --> 01:05:54.250
And we usually adopt the
convention that Fm is not
01:05:54.250 --> 01:05:55.500
equal to 0.
01:05:57.770 --> 01:05:58.090
All right?
01:05:58.090 --> 01:06:01.110
So we only write the polynomial
out to its last
01:06:01.110 --> 01:06:03.160
non-zero coefficient.
01:06:03.160 --> 01:06:09.520
In general, this could go up
to an arbitrary degree, but
01:06:09.520 --> 01:06:14.240
well, a polynomial, by
definition has a finite
01:06:14.240 --> 01:06:18.120
degree, which means it has
a finite m for which the
01:06:18.120 --> 01:06:21.590
polynomial can be written
in this way.
01:06:21.590 --> 01:06:26.710
And if the Fm is the highest
non-zero coefficient, then we
01:06:26.710 --> 01:06:31.340
say the degree of f of x is m.
01:06:35.450 --> 01:06:41.120
So all polynomials have a finite
degrees, except for 1.
01:06:41.120 --> 01:06:44.820
There is the zero polynomial,
which we have
01:06:44.820 --> 01:06:48.090
to account for somehow.
01:06:48.090 --> 01:06:51.620
And here we'll just call
it f of x equals 0.
01:06:54.270 --> 01:06:56.910
Informally, it's a polynomial,
all of those
01:06:56.910 --> 01:06:59.560
coefficients are 0.
01:06:59.560 --> 01:07:02.600
But we'll just define it
by its properties.
01:07:02.600 --> 01:07:06.280
Zero polynomial plus any other
polynomial is equal to the
01:07:06.280 --> 01:07:10.370
identity under addition
for the polynomials?
01:07:10.370 --> 01:07:12.710
What's the degree of the
zero polynomial?
01:07:17.820 --> 01:07:22.060
Anyone have a definition for
the degree of the zero
01:07:22.060 --> 01:07:23.150
polynomial?
01:07:23.150 --> 01:07:24.350
Is this well-defined?
01:07:24.350 --> 01:07:25.600
Undefined?
01:07:31.320 --> 01:07:32.030
OK.
01:07:32.030 --> 01:07:34.090
Well, I'll suggest to you
that it should be
01:07:34.090 --> 01:07:35.400
defined as minus infinity.
01:07:41.970 --> 01:07:44.630
This actually makes a lot of
things come out nicely, but it
01:07:44.630 --> 01:07:48.990
is on the other hand, you
don't have to do this.
01:07:48.990 --> 01:07:50.770
If you like, you can define
the degree of
01:07:50.770 --> 01:07:53.270
0 to be minus infinity.
01:07:53.270 --> 01:07:54.520
It's just a convention.
01:07:58.450 --> 01:07:58.650
OK.
01:07:58.650 --> 01:08:13.520
So the set of all polynomials
over F0.
01:08:13.520 --> 01:08:14.770
What's x here?
01:08:19.819 --> 01:08:22.250
I've got this thing x.
01:08:22.250 --> 01:08:23.760
What should I think
of this as being?
01:08:23.760 --> 01:08:26.880
Is this an element of a field,
or is it something else?
01:08:32.420 --> 01:08:35.800
In math, it's usually called
an indeterminate.
01:08:35.800 --> 01:08:37.180
It's just a placeholder.
01:08:37.180 --> 01:08:40.930
It's something else we stick
in order to define the
01:08:40.930 --> 01:08:41.285
polynomial.
01:08:41.285 --> 01:08:45.282
It doesn't have a value,
in principle.
01:08:45.282 --> 01:08:52.960
A comment is made in the notes
that with real and complex
01:08:52.960 --> 01:08:56.939
polynomials, you often think
of x as being a real or
01:08:56.939 --> 01:08:57.960
complex number.
01:08:57.960 --> 01:09:00.890
In other words, you evaluate the
polynomial at some alpha
01:09:00.890 --> 01:09:03.810
in the real or the complex field
by just substituting
01:09:03.810 --> 01:09:05.800
alpha for x.
01:09:05.800 --> 01:09:15.729
And in fact, two polynomials are
equal if they evaluate to
01:09:15.729 --> 01:09:18.479
the same value for all the
alphas and they're
01:09:18.479 --> 01:09:22.220
unequal if not true.
01:09:22.220 --> 01:09:24.760
When we get to finite fields,
it's important this be an
01:09:24.760 --> 01:09:26.180
indeterminate.
01:09:26.180 --> 01:09:32.120
Because consider x and x squared
as polynomials over
01:09:32.120 --> 01:09:36.939
the binary field F2.
01:09:36.939 --> 01:09:40.120
What are the values of these?
01:09:40.120 --> 01:09:48.029
We'll call this F1 of x equals
x, F2 of x equals x squared.
01:09:48.029 --> 01:09:56.320
Then F1 of 0 is 0 and
F1 of 1 is 1.
01:09:56.320 --> 01:09:56.670
Right?
01:09:56.670 --> 01:09:59.950
If I evaluate these at field
elements, the two field
01:09:59.950 --> 01:10:02.060
elements, I get 0 and 1.
01:10:02.060 --> 01:10:10.260
F2 of 0 is equal to 0, and
F2 of 1 is equal to 1.
01:10:10.260 --> 01:10:14.550
But these are not the same
polynomial, all right?
01:10:14.550 --> 01:10:20.880
So x is not to be considered
as a field element.
01:10:20.880 --> 01:10:26.180
It's to be considered just as
a placeholder, a way of
01:10:26.180 --> 01:10:27.710
holding up these polynomials.
01:10:27.710 --> 01:10:31.485
It's actually most important
in multiplication.
01:10:31.485 --> 01:10:34.835
But we gather common
terms in x.
01:10:34.835 --> 01:10:36.950
This is the multiplication
rule.
01:10:36.950 --> 01:10:38.840
But it's just something we
introduce to define the
01:10:38.840 --> 01:10:41.170
polynomial.
01:10:41.170 --> 01:10:41.510
All right.
01:10:41.510 --> 01:10:47.870
So the set of all polynomials
over F in x, or in x over F,
01:10:47.870 --> 01:10:53.800
is simply written as F
square brackets of x.
01:10:53.800 --> 01:10:55.570
That's the convention.
01:10:55.570 --> 01:10:59.320
So that's what I will write
when I mean that.
01:10:59.320 --> 01:11:05.640
And it includes all sequences
like this of finite degree,
01:11:05.640 --> 01:11:07.280
starting at 0, ending
somewhere.
01:11:09.870 --> 01:11:12.710
And also the zero polynomial.
01:11:18.560 --> 01:11:19.865
How do you add polynomials?
01:11:22.440 --> 01:11:26.970
Let's talk about the arithmetic
properties of
01:11:26.970 --> 01:11:27.730
polynomials.
01:11:27.730 --> 01:11:28.780
You know how to do this.
01:11:28.780 --> 01:11:36.230
If you have F0 plus F1 plus F2
and so forth, you have some
01:11:36.230 --> 01:11:40.320
other polynomial doesn't have
to be the same degree --
01:11:40.320 --> 01:11:44.720
G of x is G0 plus G1
x, up there --
01:11:44.720 --> 01:11:47.150
how do you add these together?
01:11:47.150 --> 01:11:49.250
Component-wise.
01:11:49.250 --> 01:11:57.850
Sum is F0 plus G0 plus
F1 plus G1 x plus 2x
01:11:57.850 --> 01:12:00.500
squared and so forth.
01:12:00.500 --> 01:12:03.170
That's an example.
01:12:03.170 --> 01:12:08.280
So you basically insert dummy
0's out here above the highest
01:12:08.280 --> 01:12:11.610
degree term in G. You add
them up component-wise.
01:12:11.610 --> 01:12:13.400
The addition operation
is where?
01:12:13.400 --> 01:12:16.080
In this field, you have addition
operation in that
01:12:16.080 --> 01:12:17.970
failed field, so you
can do this.
01:12:17.970 --> 01:12:21.150
And you get some result which
is clearly itself a
01:12:21.150 --> 01:12:22.580
polynomial.
01:12:22.580 --> 01:12:25.680
If all the coefficients are
0, you declare that
01:12:25.680 --> 01:12:27.260
the result is 0.
01:12:27.260 --> 01:12:30.360
Otherwise the result
has some degree.
01:12:30.360 --> 01:12:33.320
If you add two polynomials with
different degree, the
01:12:33.320 --> 01:12:35.400
degree of the resulting
polynomial is going to be the
01:12:35.400 --> 01:12:36.750
higher degree.
01:12:36.750 --> 01:12:39.840
If they have the same degree,
you could get cancellation in
01:12:39.840 --> 01:12:42.960
the highest order term, and get
a result which is of lower
01:12:42.960 --> 01:12:46.530
degree, all the way down to 0.
01:12:46.530 --> 01:12:47.100
All right?
01:12:47.100 --> 01:12:55.730
So addition is component-wise
the degree of the result is
01:12:55.730 --> 01:13:02.890
less than or equal to the max
degree of the components.
01:13:02.890 --> 01:13:04.445
So we do addition.
01:13:04.445 --> 01:13:05.845
How do we do multiplication?
01:13:09.130 --> 01:13:11.960
You all know how to do
polynomial multiplication.
01:13:14.550 --> 01:13:16.300
Example.
01:13:16.300 --> 01:13:24.360
F0 plus F1 of x times
G0 plus G1 of x.
01:13:24.360 --> 01:13:25.170
What do you do?
01:13:25.170 --> 01:13:27.480
You just multiply it
out term by term.
01:13:27.480 --> 01:13:43.530
F0 G0 plus F1 G0 x plus F0 G1
x plus F1 G1 x squared.
01:13:43.530 --> 01:13:45.300
You can combine these
two together.
01:13:48.620 --> 01:13:51.575
And that's your answer, which
clearly is a polynomial.
01:13:58.320 --> 01:14:01.390
So that's one way of doing it,
is multiply out term by term,
01:14:01.390 --> 01:14:03.710
collect the terms.
01:14:03.710 --> 01:14:07.970
The result of this is that what
you get is a convolution
01:14:07.970 --> 01:14:10.640
for each of the coefficients
in the new polynomial.
01:14:10.640 --> 01:14:15.980
You convolve, just by the
ordinary rules of polynomial
01:14:15.980 --> 01:14:20.840
addition, you can basically turn
this around, you convolve
01:14:20.840 --> 01:14:22.750
it, and you'll get these
coefficients.
01:14:22.750 --> 01:14:26.330
This is written out
in the notes.
01:14:26.330 --> 01:14:30.530
So we know how to do polynomial
multiplication.
01:14:30.530 --> 01:14:34.360
What are some of
its properties?
01:14:34.360 --> 01:14:36.310
How is this defined
again in F?
01:14:36.310 --> 01:14:38.370
We see we're now going to need
the multiplicative of
01:14:38.370 --> 01:14:46.110
properties of our field F.
All of these products and
01:14:46.110 --> 01:14:49.210
ultimately convolutions are
performed in F. That's why we
01:14:49.210 --> 01:14:53.070
did these coefficients to be in
a field, so we can do all
01:14:53.070 --> 01:14:55.860
these things.
01:14:55.860 --> 01:14:57.690
All right.
01:14:57.690 --> 01:14:58.940
What are some of
the properties?
01:15:02.930 --> 01:15:08.360
What is the degree of the
product of two polynomials?
01:15:08.360 --> 01:15:12.770
It's going to be the sum
of the degrees, right?
01:15:12.770 --> 01:15:16.650
Provided that both the
polynomials are not 0.
01:15:16.650 --> 01:15:20.790
The highest non-zero term is
clearly going to be a term of
01:15:20.790 --> 01:15:26.450
this kind, and it's going to be
a coefficient of x to the
01:15:26.450 --> 01:15:28.440
sum of the degrees.
01:15:28.440 --> 01:15:32.260
And since F1 and G1 are both
non-zero, by the way we write
01:15:32.260 --> 01:15:35.130
polynomials, them this
highest order term
01:15:35.130 --> 01:15:36.760
is going to be non-zero.
01:15:36.760 --> 01:15:42.080
But we also have to basically
have another rule that 0 times
01:15:42.080 --> 01:15:45.720
f of x is equal to 0.
01:15:45.720 --> 01:15:49.824
So that's the way we
multiply by 0.
01:15:49.824 --> 01:15:53.560
And how does the degree formula
work in this case?
01:15:53.560 --> 01:15:55.360
Well, this is why I defined
the degree of
01:15:55.360 --> 01:15:57.480
0 to be minus infinity.
01:15:57.480 --> 01:16:05.380
So the degree of the product
0 times f of x is --
01:16:05.380 --> 01:16:09.410
I've defined this to be the
degree of 0 plus the
01:16:09.410 --> 01:16:11.260
degree of f of x.
01:16:11.260 --> 01:16:12.130
This is finite.
01:16:12.130 --> 01:16:13.910
This is minus infinity.
01:16:13.910 --> 01:16:18.370
So the sum is minus infinity,
and so it holds.
01:16:18.370 --> 01:16:18.920
OK?
01:16:18.920 --> 01:16:20.980
That's why we defined
the degree of
01:16:20.980 --> 01:16:22.550
0 to be minus infinity.
01:16:22.550 --> 01:16:26.520
We don't have to, but it's just
so that the sum of the
01:16:26.520 --> 01:16:29.890
degrees formula continues
to work, even if we're
01:16:29.890 --> 01:16:32.140
multiplying by 0.
01:16:32.140 --> 01:16:34.550
Is there a multiplicative
identity for polynomials?
01:16:38.170 --> 01:16:38.440
Yeah.
01:16:38.440 --> 01:16:41.600
What is the multiplicative
identity?
01:16:41.600 --> 01:16:42.180
1.
01:16:42.180 --> 01:16:42.660
OK.
01:16:42.660 --> 01:16:49.175
So one times f of x is
equal to f of x.
01:16:51.690 --> 01:16:55.900
So that's one of the properties
of a field.
01:16:55.900 --> 01:16:57.450
Gee.
01:16:57.450 --> 01:17:02.920
The set of all polynomials over
F in x, is this a field?
01:17:02.920 --> 01:17:06.710
Let's go back and check
our field axioms.
01:17:06.710 --> 01:17:10.245
Under addition, does f of
x form an abelian group?
01:17:13.640 --> 01:17:14.740
Does it have the
group property?
01:17:14.740 --> 01:17:16.900
If we add two polynomials, do
we get another polynomial?
01:17:19.460 --> 01:17:24.590
Do we have cancellation, if F1
plus F2 equals F1 plus F3, is
01:17:24.590 --> 01:17:26.320
it necessarily true
that F2 equals F3?
01:17:29.440 --> 01:17:30.390
Yeah, it is.
01:17:30.390 --> 01:17:33.360
In fact, this looks very
much under addition.
01:17:36.480 --> 01:17:39.815
These look like vectors.
01:17:39.815 --> 01:17:42.960
If we confine ourselves to the
set of all polynomials of
01:17:42.960 --> 01:17:46.660
degree m or less,
we can add them.
01:17:46.660 --> 01:17:54.400
And it looks very much like the
vector space f to the m.
01:17:54.400 --> 01:17:59.130
Set of all polynomials of degree
m or less corresponds
01:17:59.130 --> 01:18:04.140
to the vector space Fm, which
does have the group property
01:18:04.140 --> 01:18:07.450
under addition.
01:18:07.450 --> 01:18:12.732
So in fact, for f of x, yes.
01:18:12.732 --> 01:18:16.830
It is an abelian group under
addition with identity being
01:18:16.830 --> 01:18:18.080
the 0 polynomial.
01:18:21.530 --> 01:18:26.030
And all of f of x is an infinite
abelian group.
01:18:26.030 --> 01:18:29.150
If we just take polynomials of
degree m or less and restrict
01:18:29.150 --> 01:18:32.060
Fp to be a finite group, then
there are only p to the m
01:18:32.060 --> 01:18:37.230
elements, and we would have
a finite abelian group.
01:18:37.230 --> 01:18:38.370
OK.
01:18:38.370 --> 01:18:38.950
Well.
01:18:38.950 --> 01:18:44.405
Are the polynomials an abelian
group under multiplication?
01:18:47.350 --> 01:18:49.780
It has an identity.
01:18:49.780 --> 01:18:54.020
It has all the arithmetic
properties you might expect.
01:18:54.020 --> 01:18:55.170
It's commutative.
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.
01:18:59.130 --> 01:19:00.380
So it's abelian.
01:19:02.610 --> 01:19:05.560
It has cancellation.
01:19:05.560 --> 01:19:08.840
f of x g of x is equal
to f of x h of x.
01:19:08.840 --> 01:19:12.520
That's true if and only if g
of x is equal to h of x.
01:19:16.040 --> 01:19:18.520
Is it missing anything?
01:19:18.520 --> 01:19:19.710
Inverse, right?
01:19:19.710 --> 01:19:21.450
Very good.
01:19:21.450 --> 01:19:22.700
Just like the integers.
01:19:22.700 --> 01:19:24.965
Not all the polynomials
have inverses.
01:19:28.610 --> 01:19:32.808
Which are the polynomials
that do have inverses?
01:19:32.808 --> 01:19:34.058
AUDIENCE: [INAUDIBLE]
01:19:37.310 --> 01:19:37.750
PROFESSOR: No.
01:19:37.750 --> 01:19:41.320
There are more than that.
01:19:41.320 --> 01:19:46.983
So now we're getting into
polynomial factorization.
01:19:51.430 --> 01:19:57.100
And the particular topic
is units, which are the
01:19:57.100 --> 01:19:58.350
invertible polynomials.
01:20:06.870 --> 01:20:08.120
And what are they?
01:20:10.294 --> 01:20:11.970
Does the 0 polynomial
have an inverse?
01:20:20.960 --> 01:20:22.210
We're a little unsure, are we?
01:20:24.836 --> 01:20:28.045
What could it possibly be?
01:20:28.045 --> 01:20:30.230
If it had an inverse,
this would mean 0
01:20:30.230 --> 01:20:34.510
times f of x is --
01:20:34.510 --> 01:20:39.450
well, 0 times f of x would have
to be a one-to-one map to
01:20:39.450 --> 01:20:42.590
all of f of x.
01:20:42.590 --> 01:20:44.280
But it isn't.
01:20:44.280 --> 01:20:46.450
It simply maps to 0.
01:20:46.450 --> 01:20:50.270
Doesn't have an inverse.
01:20:50.270 --> 01:20:55.520
What about the non-zero
polynomials
01:20:55.520 --> 01:20:56.520
that have degree 0?
01:20:56.520 --> 01:21:06.270
In other words, the degree
0 polynomial, is simply
01:21:06.270 --> 01:21:10.650
something that looks like this.
fx equals f0, where f0
01:21:10.650 --> 01:21:11.900
is not equal to 0.
01:21:16.400 --> 01:21:19.386
Is that invertible?
01:21:19.386 --> 01:21:22.600
Yeah, because f0 is in the
field, and it has an inverse.
01:21:22.600 --> 01:21:29.930
So this has inverse 1 over f of
x, if you like, equals just
01:21:29.930 --> 01:21:31.340
f0 minus 1.
01:21:31.340 --> 01:21:32.950
1 over f0.
01:21:38.880 --> 01:21:40.810
By the rules, you multiply
these two
01:21:40.810 --> 01:21:42.085
things, and you get 1.
01:21:48.630 --> 01:21:49.810
OK.
01:21:49.810 --> 01:21:53.280
So the units in this --
01:21:53.280 --> 01:21:54.710
well, I'm sorry.
01:21:54.710 --> 01:21:58.460
Take a degree 1 or a higher
polynomial --
01:21:58.460 --> 01:21:59.710
does that have an inverse?
01:22:03.640 --> 01:22:06.535
Let's suppose we take a
degree 1 polynomial --
01:22:10.680 --> 01:22:16.000
say, F0 plus F1 x --
01:22:16.000 --> 01:22:18.010
and what I want to find
is its inverse.
01:22:21.160 --> 01:22:24.010
Let's call it g of x.
01:22:24.010 --> 01:22:26.610
Is it possible to find a g of
x such that the product of
01:22:26.610 --> 01:22:27.860
these two is 1?
01:22:35.400 --> 01:22:39.170
Clearly not, because by our
degree rule, what is the
01:22:39.170 --> 01:22:41.850
degree of this product
going to be?
01:22:41.850 --> 01:22:45.720
The degree is going to be the
sum of this degree plus this
01:22:45.720 --> 01:22:50.200
degree, the degree of f plus
the degree of g, which is
01:22:50.200 --> 01:22:53.850
going to have to
be at least 1.
01:22:53.850 --> 01:22:56.780
Provided that g of x is not 0,
but clearly g of x is not the
01:22:56.780 --> 01:22:59.600
solution we're looking
for here, either.
01:22:59.600 --> 01:23:08.910
So this can't be true, and the
invertible polynomials are the
01:23:08.910 --> 01:23:17.850
degree 0 polynomials, which
means they're basically the
01:23:17.850 --> 01:23:24.050
non-zero elements of F. Yes.
01:23:24.050 --> 01:23:25.150
Considered as polynomials.
01:23:25.150 --> 01:23:26.785
AUDIENCE: And how are the
[UNINTELLIGIBLE PHRASE]?
01:23:33.000 --> 01:23:33.690
PROFESSOR: Yes, indeed.
01:23:33.690 --> 01:23:34.940
AUDIENCE: [INAUDIBLE]
01:23:37.920 --> 01:23:38.610
PROFESSOR: Ah, no.
01:23:38.610 --> 01:23:43.560
We haven't introduced modulo
polynomials and right.
01:23:48.010 --> 01:23:53.480
The powers and the indices are
the integers from 0 up to some
01:23:53.480 --> 01:23:54.730
finite number.
01:23:57.170 --> 01:23:58.160
OK.
01:23:58.160 --> 01:23:59.790
It's time to quit.
01:23:59.790 --> 01:24:02.750
We'll finish this.
01:24:02.750 --> 01:24:06.400
We'll, I believe, to be able
to certainly finish chapter
01:24:06.400 --> 01:24:10.140
seven, maybe get a little bit
into chapter eight, next time,
01:24:10.140 --> 01:24:12.945
on Wednesday.
01:24:12.945 --> 01:24:14.195
And we'll see you then.