WEBVTT 00:00:00.920 --> 00:00:05.940 PROFESSOR: So we're slightly into chapter seven, which is 00:00:05.940 --> 00:00:08.029 the algebra chapter. 00:00:08.029 --> 00:00:12.600 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. 00:00:16.970 --> 00:00:21.700 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 00:00:30.780 --> 00:00:34.470 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. 00:00:42.380 --> 00:00:46.750 I'm trying to give you a short course in algebra, really, in 00:00:46.750 --> 00:00:49.200 two lectures or fewer. 00:00:49.200 --> 00:00:53.100 And clearly I'm going to miss a lot of things. 00:00:53.100 --> 00:00:55.155 In particular, I'm not going to cover even everything 00:00:55.155 --> 00:00:58.830 that's in chapter seven, which itself is a highly compressed 00:00:58.830 --> 00:01:01.950 introduction to finite fields. 00:01:01.950 --> 00:01:05.660 I'm trying to do this while remaining faithful to the 00:01:05.660 --> 00:01:09.020 philosophy of this course and other courses at MIT, which is 00:01:09.020 --> 00:01:12.880 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. 00:01:15.810 --> 00:01:18.970 So it's a little tough and it forces me to go a little fast, 00:01:18.970 --> 00:01:22.820 but I hope that you can keep up, and especially with the 00:01:22.820 --> 00:01:26.680 assistance of the notes or the many possible other things you 00:01:26.680 --> 00:01:29.680 could read on this subject, which are listed in the notes, 00:01:29.680 --> 00:01:31.390 you'll be able to keep up. 00:01:31.390 --> 00:01:33.960 And some of you, of course, have seen this in other places 00:01:33.960 --> 00:01:37.490 in more extended form. 00:01:37.490 --> 00:01:41.340 Now, this will get us in a position to start to talk 00:01:41.340 --> 00:01:46.150 about Reed-Solomon codes, which are the single major 00:01:46.150 --> 00:01:50.250 accomplishment of the field of algebraic coding theory. 00:01:50.250 --> 00:01:53.920 Certainly for getting to capacity on the additive white 00:01:53.920 --> 00:01:58.835 Gaussian noise channel and for lots of other things, they're 00:01:58.835 --> 00:02:04.340 an extremely useful and widely implemented class of codes. 00:02:04.340 --> 00:02:06.790 And we'll be able to maybe just get to the beginning of 00:02:06.790 --> 00:02:08.910 that by Wednesday. 00:02:08.910 --> 00:02:11.610 And then I won't be here again next week, but fortunately we 00:02:11.610 --> 00:02:17.760 have an expert on campus who is far more expert than I in 00:02:17.760 --> 00:02:21.180 Reed-Solomon codes, their decoding algorithms, who has 00:02:21.180 --> 00:02:25.360 agreed to talk for two lectures, maybe one more 00:02:25.360 --> 00:02:29.570 focused on Reed-Solomon codes and one, I hope, on his whole 00:02:29.570 --> 00:02:31.040 philosophy of life. 00:02:31.040 --> 00:02:31.580 Perhaps. 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 00:02:33.830 --> 00:02:36.710 TV when I get back. 00:02:36.710 --> 00:02:40.740 Anyway, Ralf Koetter will be lecturer for Monday and 00:02:40.740 --> 00:02:42.900 Wednesday next week. 00:02:42.900 --> 00:02:46.880 I think you'll enjoy the change of pace. 00:02:46.880 --> 00:02:47.360 OK. 00:02:47.360 --> 00:02:49.780 So where are we in chapter seven? 00:02:49.780 --> 00:02:50.800 We're not very far. 00:02:50.800 --> 00:02:54.250 We're talking about these various algebraic objects. 00:02:54.250 --> 00:02:59.030 We've started with integers just to get you 00:02:59.030 --> 00:02:59.860 into the feel of it. 00:02:59.860 --> 00:03:04.260 We mainly talked about integer factorization, the Euclidean 00:03:04.260 --> 00:03:06.860 division algorithm, things that you've known for a very 00:03:06.860 --> 00:03:11.640 long time, basically here because A, we're going to be 00:03:11.640 --> 00:03:15.100 using integers as we go along, and their factorization 00:03:15.100 --> 00:03:18.360 properties, B, it's a model for polynomials, which behave 00:03:18.360 --> 00:03:22.080 very much the same way as integers because they're both 00:03:22.080 --> 00:03:23.390 principal ideal domains. 00:03:23.390 --> 00:03:28.380 In particular, we looked at the integers mod n with the 00:03:28.380 --> 00:03:34.900 rules of mod n arithmetic, which we're going to call Zn. 00:03:34.900 --> 00:03:38.800 This is simply 0 through n minus 1 with the mod n 00:03:38.800 --> 00:03:41.440 arithmetic rules. 00:03:41.440 --> 00:03:44.920 And then we went on to groups. 00:03:44.920 --> 00:03:49.140 We first gave the standard axioms for groups, and then I 00:03:49.140 --> 00:03:53.000 gave you an alternative set of axioms which focused on this 00:03:53.000 --> 00:03:54.370 permutation property. 00:03:54.370 --> 00:03:59.590 If you add, I'm calling the group operation addition, 00:03:59.590 --> 00:04:03.450 because essentially all the groups we talk about are going 00:04:03.450 --> 00:04:05.240 to be abelian -- 00:04:05.240 --> 00:04:08.950 if we add a group element to the group, 00:04:08.950 --> 00:04:10.530 what do we get back? 00:04:10.530 --> 00:04:12.800 We get the whole group again. 00:04:12.800 --> 00:04:15.040 It's permuted, it's the entire group, it's 00:04:15.040 --> 00:04:16.589 in a different order. 00:04:16.589 --> 00:04:18.050 All right? 00:04:18.050 --> 00:04:21.610 And with this and the identity, this plus the 00:04:21.610 --> 00:04:26.770 identity and the associativity axiom are also a sufficient 00:04:26.770 --> 00:04:28.960 set of axioms for the group. 00:04:28.960 --> 00:04:32.010 And I think this is really the most useful thing to think 00:04:32.010 --> 00:04:33.540 about with a group. 00:04:33.540 --> 00:04:35.460 We've also called it the group property when we 00:04:35.460 --> 00:04:36.845 talked about codes. 00:04:36.845 --> 00:04:40.000 You know, if you add the code word to all the elements in 00:04:40.000 --> 00:04:41.440 the code, you get the code back. 00:04:44.190 --> 00:04:47.090 And you saw how useful that was for seeing certain 00:04:47.090 --> 00:04:51.110 symmetry properties of minor linear block codes. 00:04:51.110 --> 00:04:54.150 So we even talked about cyclic groups, specifically finite 00:04:54.150 --> 00:04:55.570 cyclic groups. 00:04:55.570 --> 00:04:59.340 And we showed that all them basically are 00:04:59.340 --> 00:05:01.600 isomorphic to z mod n. 00:05:04.305 --> 00:05:06.800 A cyclic group is defined by a single generator. 00:05:06.800 --> 00:05:10.830 If we identify that generator with one, G plus G is 00:05:10.830 --> 00:05:14.070 identified with 2, and so forth, then we get an addition 00:05:14.070 --> 00:05:16.840 table which is exactly the same addition table as Zn. 00:05:16.840 --> 00:05:19.770 And that's what we mean when we say two groups are 00:05:19.770 --> 00:05:20.480 isomorphic. 00:05:20.480 --> 00:05:23.680 So all finite cyclic groups look like Zn. 00:05:23.680 --> 00:05:26.900 You can think of them as being images of Zn. 00:05:26.900 --> 00:05:28.400 All right? 00:05:28.400 --> 00:05:30.930 It's the only one you need to know about. 00:05:30.930 --> 00:05:31.500 OK. 00:05:31.500 --> 00:05:37.950 So that's where we are any questions on this material? 00:05:37.950 --> 00:05:41.110 Pretty easy, I think. 00:05:41.110 --> 00:05:43.310 Terribly easy if you've ever seen any of this before. 00:05:43.310 --> 00:05:48.080 Probably takes a little absorbing if you haven't. 00:05:48.080 --> 00:05:48.330 OK. 00:05:48.330 --> 00:05:53.880 Now the next natural subject to talk about is subgroups. 00:05:53.880 --> 00:05:55.950 And what is a subgroup? 00:05:55.950 --> 00:06:00.640 A subgroup simply a subset of elements in the group which, 00:06:00.640 --> 00:06:04.680 together with the group operation already specified in 00:06:04.680 --> 00:06:10.345 G, which we're calling circle plus, is itself a group. 00:06:12.870 --> 00:06:13.810 What does that mean? 00:06:13.810 --> 00:06:17.220 Well, associativity comes for free, because we already have 00:06:17.220 --> 00:06:21.840 that property for circle plus in G. Obviously H has to 00:06:21.840 --> 00:06:28.460 include the identity in order to satisfy the group axioms. 00:06:28.460 --> 00:06:31.290 And finally, the third group axiom is this permutation or 00:06:31.290 --> 00:06:34.770 group property that if we add any element of the subgroup to 00:06:34.770 --> 00:06:38.430 itself, we have to stay within the subgroup and ultimately 00:06:38.430 --> 00:06:40.270 generate the whole subgroup. 00:06:40.270 --> 00:06:43.150 That means that subtraction, 00:06:43.150 --> 00:06:46.990 cancellation hold in the subgroup. 00:06:46.990 --> 00:06:48.310 All right? 00:06:48.310 --> 00:06:51.390 So that's clear. 00:06:51.390 --> 00:06:55.460 What's an example of a subgroup? 00:06:55.460 --> 00:06:57.180 If we have -- 00:06:57.180 --> 00:07:01.180 we talked about Z10 as the group. 00:07:01.180 --> 00:07:02.310 What would be a subgroup? 00:07:02.310 --> 00:07:03.560 Anybody? 00:07:08.167 --> 00:07:10.120 AUDIENCE: [INAUDIBLE] 00:07:10.120 --> 00:07:12.220 PROFESSOR: 0 to 4. 00:07:12.220 --> 00:07:12.750 OK. 00:07:12.750 --> 00:07:18.970 So you're proposing that H is the elements 0, 1, 2, 3, 4 out 00:07:18.970 --> 00:07:22.730 of G. Does that work? 00:07:22.730 --> 00:07:25.670 This doesn't include 0. 00:07:25.670 --> 00:07:27.110 But suppose I add 3 and 4? 00:07:30.100 --> 00:07:34.950 AUDIENCE: I assume that you have modules for a reason. 00:07:34.950 --> 00:07:35.650 PROFESSOR: No. 00:07:35.650 --> 00:07:41.370 In this group, the group operations modulo 10 addition. 00:07:41.370 --> 00:07:43.720 Subgroup has to have the same operation as the 00:07:43.720 --> 00:07:44.970 group it came from. 00:07:48.600 --> 00:07:53.840 What you've got here is you've already got, in essence, a 00:07:53.840 --> 00:07:57.036 quotient group. 00:07:57.036 --> 00:07:58.630 Or at least that's where you're headed. 00:07:58.630 --> 00:08:01.130 So this is not a subgroup. 00:08:01.130 --> 00:08:02.300 Thank you for the suggestion. 00:08:02.300 --> 00:08:03.310 Fails. 00:08:03.310 --> 00:08:03.970 Anyone else? 00:08:03.970 --> 00:08:04.300 What? 00:08:04.300 --> 00:08:05.250 AUDIENCE: 0,1? 00:08:05.250 --> 00:08:07.020 PROFESSOR: 0,1. 00:08:07.020 --> 00:08:07.230 OK. 00:08:07.230 --> 00:08:10.610 Let's try that. 00:08:10.610 --> 00:08:13.990 And it contains the identity 0 plus 1 is 00:08:13.990 --> 00:08:14.700 certainly in the group. 00:08:14.700 --> 00:08:17.160 How about 1 plus 1? 00:08:17.160 --> 00:08:20.810 It's not in the group. 00:08:20.810 --> 00:08:21.617 0 and 5? 00:08:21.617 --> 00:08:22.867 That sounds more promising. 00:08:28.910 --> 00:08:31.580 Now, 0 plus -- what's the addition table of this? 00:08:34.419 --> 00:08:35.669 0, 5, 0, 5, 0, 5. 00:08:38.780 --> 00:08:39.940 What's 5 plus 5? 00:08:39.940 --> 00:08:40.570 It's 10. 00:08:40.570 --> 00:08:43.360 But mod-10 , that's 0. 00:08:43.360 --> 00:08:45.290 So it seems we do have a group. 00:08:45.290 --> 00:08:51.750 And in fact, this is a finite cyclic group generated by 5, 00:08:51.750 --> 00:08:57.500 and has two elements, so it's isomorphic to Z2. 00:08:57.500 --> 00:08:59.740 In other words, the addition table looks just like the 00:08:59.740 --> 00:09:01.600 addition table of Z2 with a relabeling. 00:09:05.500 --> 00:09:06.860 OK. 00:09:06.860 --> 00:09:10.080 Any other subgroups of Z10? 00:09:10.080 --> 00:09:11.280 The even integers. 00:09:11.280 --> 00:09:11.680 There. 00:09:11.680 --> 00:09:14.010 Now we're really smoking. 00:09:14.010 --> 00:09:18.890 H equals 0,2,4,6,8. 00:09:18.890 --> 00:09:20.580 Those are all the even integers in Z10. 00:09:23.330 --> 00:09:26.380 And again, evens plus evens equal evens. 00:09:26.380 --> 00:09:31.880 So we get a group of five elements, satisfies is the 00:09:31.880 --> 00:09:39.320 group property, and it's isomorphic to what? 00:09:39.320 --> 00:09:39.845 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 00:09:44.490 --> 00:09:45.740 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. 00:10:28.420 --> 00:10:29.670 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.