## MATH 113 Introduction to Abstract Algebra (Fall 2003):

Instructor: Alexander Yong, 1035 Evans Hall, ayong@math.berkeley.edu

Lectures: Tuesdays and Thursdays, 3:30-5:00pm, Room 75 Evans

Office hours: Tu 2:00-3:00pm, Th 5:00-6:00pm. There will also be a GSI ( Jameel Al-Aidroos) available Wednesdays 12:10-5:00PM and Thursdays 9:10AM-2:00PM in 891 Evans.

Required Text: John B. Fraleigh, "A First Course in Abstract Algebra, 7th Edition", Addison-Wesley, 2003.

Other recommended texts:

1. Joseph J. Rotman, "A First Course in Abstract Algebra, 2nd Edition", Prentice Hall, 2000.

2. Michel Artin, "Algebra", Prentice Hall, 1991.

3. I.N. Herstein, "Abstract Algebra, 3rd edition", Prentice Hall, 1996.

Grading: homework (30%), Midterm 1 (15%), Midterm 2 *or* essay on an aspect of Grobner bases (15%) [due **December 10**, put under my office door], final exam (40%)

Homework: Assignments will be posted weekly starting September 2nd on the course webpage and are due in class the following week.

Homework 0: Send me an email about you and your (mathematical) background so I can get to better know you.

Exams: There will be two in class midterms; the first will be on October 14th, covering group theory), the second on Tuesday, November 25th (**note change in date**) covering ring theory and some field theory. The final exam will be cumulative, it will be on **Wednesday, December 17, 2003** from **12:30PM - 3:30 PM** in **3 Evans Hall**.

Prerequisites: Math 54 or equivalent knowledge of linear algebra.

Syllabus: (From the course catalog): "Sets and relations. The integers, congruences and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions." -- as given in (parts of) chapters 0-10 of the textbook.

Outline: I will spend ~1 week on "preliminaries" (chapter 0), 5.5 weeks on "group theory" (chapters 1-3,7), 4.5 weeks on "rings and polynomials" (chapters 4-5) and 4 weeks on "elements of field theory" (chapters 6,9-10). Of course there will not be time to cover **every** aspect of these chapters!

Part I: Preliminaries:

Lecture 1: Why you might like abstract algebra, first day survey (PDF file) , GL_n(R), two ways to look at S_n (but why are they the same??), Fomin and Greene's problem.

Part II: Group theory:

Lecture 2: Definition of a group, homomorphisms, isomorphisms, basic properties of groups, finite group tables, subgroups, more examples of groups, including the automorphism group of a finite graph.

Lecture 3: More on subgroups, especially examples. Cyclic groups, any subgroup of a cyclic group is cyclic (idea of proof: in the end, consider the division algorithm). Assignment 1 (PDF file) handed out.

Lecture 4: Cayley's theorem (idea of proof: any row of the group table of G is a permutation of G), introduction to group actions: orbits, cycles of permutations. Equivalence relations.

Lecture 5: Sign of a permutation, alternating groups, left and right cosets ("cells" of an equivalence relation), Lagrange's theorem that if H is a subgroup of G, then order(H) divides order(G) (idea of proof: any subgroup H has the same number of elements as any left coset aH). Debate about induction: induction *does prove* that 1+2+...+n = n(n+1)/2, but it hides *why* that statement is true; the debate continues...

Lecture 6: Direct products and finitely generated abelian groups. Z_n x Z_m is cyclic when gcd(m,n)=1 (proof: if gcd(m,n)=1 then (1,1) generates, if gcd(m,n)>1 then any element (a,b) added to itself lcm(m,n) times is zero). The Fundamental theorem of finitely generated abelian groups. Assignment 1 collected and Assignment 2 (PDF file) handed out, due next Thursday.

Lecture 7: More on homomorphisms, normal subgroups and factor groups. A factor group G/H is actually a group if and only if H is normal. Normal subgroups come from kernels of homomorphisms-- for free we get that the alternating group A_n is a normal subgroup of S_n (because of the sign: S_n -> {-1,1} homomorphism). How's it free? Exercise: prove the same fact from the definitions. Thus normal subgroups come up in two ways: (1) as kernels of homomorphisms and (2) as the ``right'' subgroups to look at if you want G/H to be a group. What do (1) and (2) have to do with one another? Answer: the fundamental homomorphism theorem.

Lecture 8: More on normal subgroups, equivalent formulations of what it means for a subgroup to be normal. The fundamental homomorphism theorem (basic conclusion: if p:G->G' is a surjective homomorphism, then G/ker(p) is isomorphic to G'). Assignment 2 collected and Assignment 3 (PDF file) handed out, due next Thursday.

Lecture 9: Inner automorphisms. Concrete calculations with normal subgroups and the fundamental homomorphism theorem. Converse of Lagranges' theorem is false. Simple groups. Homomorphisms preserve normality. Assignment 1 returned. Here are **partial** solutions to Assignment 1 regarding some of the harder problems that were graded. Please fill in any details as necessary in your own study of these questions.

Lecture 10: Maximal normal subgroups. Centers, commutator subgroups, we care since they give measures of how abelian a group is. Understanding groups, by moding out by e.g., commutator subgroups, group actions, example of group actions, actions of groups on themselves by left multiplication, conjugation. Assignment 3 collected and Assignment 4 (PDF file) handed out, due next Thursday. Here are **partial** solutions to Assignment 2 regarding some of the harder problems that were graded.

Lecture 11: Here's something I wrote about examples and how to find proofs. Since this class is the first "proof"-class for many participants, I hope it will be of use to some of you. Comments are welcome. More on group actions. Main result: a G-set X provides a homomorphism Phi:G->S_{X} (the group of permutations of X). The kernel of Phi is Stab_{G}(X), the stabilizer subgroup of X. This kernel is {e} iff Phi is 1-1 iff Stab_{G}(X)={e} iff G's action on X is "faithful". We also discussed the dihederal group D_4 in detail ("symmetries of a square"--it acts on the square), and used it to explain isotropy subgroups G_{x}, orbits Gx and the set X_{g}.

Lecture 12: Finish up on group actions. Applications of groups actions to counting--Burnside's formula (the number of orbits times the order of a group equals the sum over all |X_{g}|), which we use to count e.g., colorings of regular polygons. Assignment 4 collected and Assignment 5 (PDF file) handed out, due next Thursday.

Lecture 13: Here are **partial** solutions to Assignment 3 regarding some of the harder problems that were graded. Here's another installment of the "how to find proofs series", based on Thursday's office hour discussion. Again, I hope that these notes will be of value to some of you. Comments are welcome! This week's focus is on "finding the statement of claim". Some advanced topics: Isomorphism theorems, first (if Phi maps G onto H, then G/N is isomorphic to H where N is the kernel of Phi), second (if H is a subgroup of G and N is normal in G, then (HN)/N is isomorphic to H/(H intersect N) -- warning HN is *not* HxN, see page 308 of the text), third (if H and K are normal subgroups of G and K is a subgroup of H, then the ``ratio formula'' holds: G/H is isomorphic to (G/K)/(H/K)).

Lecture 14: The proof of Cauchy's theorem via Burnside's formula. P-(sub)groups, Sylow's 1st, 2nd and 3rd theorems. Applications. Here are **partial** solutions to Assignment 4 regarding some of the harder problems that were graded. Assignment 5 collected, no assignment this week, *midterm* October 14, in class, starting at 3:40 pm. This midterm will cover up to Lecture 12 (Assignment 5). Here is a practice midterm , which represents the approximate length of the real thing (but not necessarily which areas the questions on the real thing will come from!). You may wish to consult your text for plenty of other nice problems. **I will have a review session from 5-6pm in 1015 Evans and extra office hours (in 1035 Evans) 6:00-7:00pm. If you cannot make these times and would like to talk to me before the exam, please email me to set up an appointment before the exam. Note I will be out of town this Friday, but available Monday and Tuesday.**

Lecture 15: No lecture today. Instead we had the following midterm . We will grade the midterm out of 40 (5 free marks).

Part III: Ring theory:

Lecture 16: Introduction to rings and fields. Examples, homomorphisms, isomorphisms, division rings, integral domains. Assignment 6 (PDF file) handed out, due next Thursday.

Lecture 17: Characteristic of a ring. Fermat's Little theorem and Euler's generalization (proofs: for Fermat use Z_{p} and observe that 1,2,..,p-1 is a group under multiplication, of order p-1. Now use the fact that for any group G, x^{order(G)}=identity. So if p does not divide a, then it follows a^{p-1} is congruent to 1 modulo p. For Euler's theorem, use the group G_n={positive integers, smaller than n and relatively prime to it, under multiplication} and use the same idea). Assignment 6 was handed back, as was the midterm.

Each participant has a *choice* of either a second midterm, or an essay on some aspect of the beautiful subject of Grobner bases. If you choose the latter, I recommend that you start with the section on this subject in the textbook. Another great reference is Cox, Little and O'Shea's "Ideals, Varieties and Algorithms". I expect about a 10 page typed essay introducing Grobner bases and some basic facts about them. Include examples, and some discussion of the basic theorems about these bases, as well as a discussion of why these bases are studied. If there is a particular aspect that you find especially interesting, feel free to write about that. Try to link your essay to topics that we study in class.

Lecture 18: The field of quotients of an integral domain (e.g., extending the integers to the rationals). It was a long construction, starting from a commutative integral domain (where was commutativity used? integral domain?) doing four things:

(1) defining the field of quotients (basically a/b, where and b are in our domain D and b isn't zero, and having an equivalent relation that a/b~c/d if ad=bc)

(2) defining +, * just as you would for ordinary fractions

(3) checking that (1) and (2) give a field

(4) checking D embeds into F

The only really tricky conceptual issue about the construction was that you have to think of each element of F as an equivalence class of fractions (the analogy is to twist your usual conception and think of each "point" of the rationals as a set of equivalent fractions). Of course, that is often the confusing thing in this class: to think of a set as being a "point" of a space. Assignment 7 (PDF file) handed out, due **Tuesday November 4**. Here are the partial solutions to the first midterm. (PDF file). Here are **partial** solutions to Assignment 5 and here are **partial** solutions to Assignment 6, regarding some of the harder problems that were graded.

Lecture 19: Polynomial rings and factorizations. Evaluation homomorphisms. The division algorithm, factor theorem, Eisenstein criterion. More on homomorphisms. Definition of an ideal, factor rings. The main point is that there is a direct analogy with homomorphisms, normal subgroups, and factor groups in *group theory*. In ring theory, we require that our homomorphisms preserve both the additive and multiplicative structure. Then as a consequence, they send subrings to subrings, additive and multiplicative identities to additive and multiplicative identities in the image. Note further that since every ring R are assumed to be abelian groups under the + operation, automatically any sub*group* I or R is a *normal* subgroup. We would like to put a *ring* structure on the factor *group* R/I (i.e., we need multiplication of cosets to make sense). What do we demand of I? The answer is that I must be an *ideal*. Thus ideals play the crucial role in ring theory that normal subgroups play in group theory. They also arise in a similar way: they are kernels of ring homomorphisms, and there is a fundamental *ring* isomorphism theorem. There is ***no class Thursday*** as I'll be away . Consequently, Assignment 7 is due next Tuesday November 4th. There will also be no assignment this week to give time to prepare for the Grobner basis essay (which we now have all the basic background material for, see Section 28, page 254 of the text). I will have office hours Wednesday 5-6pm.

Lecture 20: More on ideals. We reviewed the basic yoga of abstract algebra by explaining why ideals are ring theory analogues of normal subgroups. Then explained the ring theory analogues of homomorphism theorems: an subset S of a ring R is an ideal if and only if it is the kernel of a ring homomorphism and if and only if R/S is a ring. The fundamental theorem of ring homomorphisms. The basic point was that "half" of these results follow from the group theory analogue: since rings are abelian groups under +. We talked about prime and maximal ideals. The basic theorems here are that is S in R is a maximal ideal, then R/S is a field; and if S is prime, then R/S is an integral domain.

Lecture 21: Finished up on prime and maximal ideals. Talked about principal ideals. We applied what we learned to ideals in F[x]= ring of polynomials with coefficients in a field F. Theorems there are that **every** ideal is principal. This makes is easy to describe what maximal ideals look like: they are the ones generated by an irreducible polynomial. Our eventual goal is to show that every polynomial f(x) in F[x] has a root in some field E that contains F. In fact, we will define E:=F[x]/(f(x)). What remains to show is that E is a field and to explain why it does the job. Here is Assignment 8 (PDF file) , due Thursday November 13. **NOTE: the final exam date has been set (see above, under "Exams").** Here are **partial** solutions to Assignment 7, regarding some of the harder problems that were graded.

Lecture 22: Talked more about Kronecker's theorem: every non constant polynomial has a zero in some extension field. Gave some examples (including the complex numbers). Having proven this goal of ours, we turn to the classical result showing that some geometric constructions by ruler and compass are impossible. We will need to definitions and tools to get to this result. We began to talk about algebraic and transcendential numbers. We proved that if a in an extension field E of F satisfies f(a)=0 for some f in F[x]. Here is Assignment 9 (PDF file) , due Thursday November 20. There will be a substitute Tuesday, as as I'll be away .

Lecture 23: The guest lecturer is Professor Kevin Hare of UC Berkeley. Our next major goal will be to prove that a number of geometric constructions by (unmarked) ruler and compass are impossible (e.g., trisecting an arbitrary angle). To achieve this end, we will need to develop more field theory. More about extension fields was discussed; algebraic numbers, transcendential numbers, THE irreducible polynomial of an algebraic element. Also discussed was the smallest field extension F(alpha) of a field F that contains a given element alpha. The second midterm is coming up next week. It will cover the material on rings and fields up to the material from this lecture (but not any group theory material). ***You may take the midterm and decide after receiving your grade if you would like to submit an essay.***

Lecture 24: Algebraic extensions, vector spaces. No assignment this week. Midterm II, to be given on **Tuesday**, November 25 will cover the material on Rings and Fields covered up to and including the lecture of November 13. This includes Part IV, Sections 18-23; Part V, Sections 26-27, 29. **There will be a review session on Monday, November 24th. It will start in the common room lounge (Evans 1015) at 5pm-6pm. Then it will continue in my office from 6pm-??pm.** Here are **partial** solutions to Assignment 8, regarding some of the harder problems that were graded.

Lecture 25: No lecture today. Instead here is Midterm II, that was held today (Novemeber 25th, 2003) and here are the solutions to Midterm II. Please email me if you could like to know your grade (right away). The rough grade scale will be A: 38+; B: 29-37; C: 20-28; D: 11-19; E: 0-10. The essay, if you choose to do it, is due **December 10** (you may slip it under my office door).

Lecture 26: Guest Lecturer Tony Chiang will talk about algebraic extensions. An algebraic extension E of a field F is one where every element is algebraic over F. A very important theorem that will be discussed will be the following counting result: if K is a finite dimensional vector space over a field L (i.e., a "finite extension field of L) and L is a finite dimensional vector space over a field F then K is a finite dimensional vector space over F. Moreover, let [K:L] denote the vector space dimension of K over L and similarly define [L:F] and [K:F]. Then [K:F]=[K:L][L:F]. Moreover, the **proof** of this theorem is as important as the statement, in this case. The idea is that is {a_1,a_2,..,a_n} is a basis for K over L and {b_1,b_2,...,b_m} is a basis for L over F, then the mn products {a_i.b_j} is a basis for K over F (which, incidently, is what you might hope is true, and is). Needless to say, this theorem has a number of important applications. Note that the essay, if you choose to do it, is due **December 10th**. ***Additional office hours to be held by Jameel: Tuesday Dec. 9, 10:00-1:00; Wednesday Dec. 10, 12:00-4:00; and Monday, Dec. 15, 10:00-1:00.***

Lecture 27: We concluded the course with a discussion of what real numbers can be constructed by straightedge and compass (answer, all numbers that can be obtained by applying plus, minus, times, division ans square root to the number 1, a finite number of times). From this it follows that if a is constructible, then [Q(a):Q] is an integral power of 2. from this fact we deduce that various classical constructions ("doubling the square, trisecting an **arbitrary** angle) are impossible. The point is always the same: we deduce that if for some particularly well chosen case one could carry out the operation, that would amount to a construction of a certain number a. But then we prove that a satisfies an irreducible polynomial (in Q) of of degree not equal to a power of 2, a contradiction. **I will be holding my regular office hours. We will also have a review session on Monday December 15th in 1015 Evans from 4-??PM.**

Final exam: Here is the final exam, given on December 17, 2003 Final Exam, that was held today.