Math 113, Introduction to Abstract Algebra

Spring 2012

MWF 2-3 PM, 3 Evans

Welcome to the course webpage for Math 113 (Section 1). Here you will find some general info about the course. This is also the place to look for homework assignments, occasional course notes, and other things that might interest you. This webpage is also available through our course page on bspace.

Course Syllabus

• Wed 8:00-11:00AM in 939 Evans
• Wed 3:00-5:00PM in 961 Evans
• Wed 5:00-6:00PM in 736 Evans
• Thurs 2:00-6:00PM in 736 Evans

The textbook we'll be using is Fraleigh's First Course in Algebra (7th edition) -- it is nicely organized and easy to read. However, no book is a perfect fit for everyone, and there are many other books you can look at for reference (I would be more than happy to recommend one for you).

The official prerequisite for this course is MAT 54. While we won't use too much of what you may have learned in that course, the assumption is that taking MAT 54 will have equipped you with the appropriate mathematical maturity needed for this class.

Your grade for the course is determined as follows: 15% for homework, 25% for each of two midterms, 55% for the final, and -20% for your lowest exam score. There will be no make up exams.

Homework (along with occasional supplementary notes) will be posted here and will be due every Friday at 3PM in class, or in my office by the same day/time (if you choose the office option, you can slide it under the door). The lowest two homework scores will be dropped. Therefore the policy is that no late homework will be accepted, especially since we will sometimes discuss the solutions to the homework problems in class.

The midterms will be in class on Wednesday, February 29 and Wednesday, April 4.

Abstract Algebra (a.k.a. just Algebra) is a beautiful subject which will arm you with tools that will prove indispensable in a very broad range of mathematics courses (from geometry to number theory) in the future. By the end of the semester, we will cover the basics of groups, rings, and fields. Basically, these are all sets which come together with certain operations: a group is a set with one binary operation which abides by some rules; a ring is a set with two binary operations which abide by some rules; and a field is a kind of special ring.

For many students this course may be the first (or one of the first) course in which they are challenged to think like a mathematician: through homeworks, exams, and hopefully discussions with classmates, students will become comfortable with creating and writing rigorous mathematical proofs. This skill is essential to all upper division courses in mathematics, and to any mathematician in general.

• Preliminaries: overview of sets, relations, functions, and proof techniques. Modular arithmetic and some basic number theory. Reading: Section 0 of textbook , M. Hutchings' notes on proofs, and my notes on basic number theory.
• Groups: permutation groups, symmetry groups, and many other examples of both finite and infinite groups. Reading: Chapters 1, 2, 3, possibly a bit of 7 (although Chapter 7 is more within the scope of Math 114 than 113).
• Rings: various examples, focusing heavily on rings of polynomials, factorization of polynomials, ideals, and more. Reading: Chapters 4, 5, and 9.
• Fields: algebraic extensions, finite fields, and why you can't trisect a 60 degree angle with a straight edge and compass even if you try very hard. Reading: Chapter 6. The avid student might be interested to continue by reading Chapter 10, or taking Math 114, since it is unlikely we will have time for it in this class.

• 1/18, 1/20: Basic number theory: division algorithm, greatest common divisors, euclidean algorithm, solving linear congruences. Reading: everything except the last section of notes on number theory
• 1/23, 1/25, 1/27: Fundamental theorem of arithmetic; Sets and such: unions, intersections, De Morgan's laws, cardinality, surjections, injections, equivalence relations. Binary operations, isomorphic binary structures, and introduction to groups. Reading: Sections 0, 2, 3, beginning of section 4, and the last section of notes on number theory.
• 1/30, 2/1: Groups, properties of groups and examples: matrix groups, Z/nZ, permutation groups, dihedral groups, and more. Quite a few of the examples we'll discuss aren't really in the book. Reading: Section 4, notes on dihedral groups posted below.
• 2/3: Subgroups and examples thereof. Reading: Section 5.
• 2/6: Cyclic groups and their subgroups. A quick mention of subgroups generated by a subset of a group G. Reading: Section 6 and just a bit of Section 7 (note that we skipped most of Section 7).
• 2/8, 2/10: Permutations galore: Cayley's theorem, permutations as products of disjoint cycles, permutations as products of transpositions, even/odd permutations, the alternating group. Reading: Sections 8 and 9.
• 2/13: Finishing permutations. Reading: end of Section 9.
• 2/15: Cosets and Lagrange's theorem. Reading: Section 10.
• 2/17: Direct products. Fundamental theorem of finitely generated abelian groups. Reading: Section 11. Note we skip Section 12 completely, even though it is a stimulating read.
• 2/22: Homomorphisms, kernels, etc. Reading: Section 13.
• 2/24: Conjugation, normal subgroups, quotient groups (a.k.a. factor groups). Reading: Section 14.
• 2/27: More on quotient groups and normal subgroups, the fundamental homomorphism theorem. Reading: Sections 14 and 15.
• 2/29: Midterm 1 in 60 Evans. Covers all material covered in class and homework through 2/22. Midterm 1 solutions. The mean was 38/50.
• 3/2: More quotient group computations. Commutator subgroup and abelianization. Discussion of simple groups. Reading: Section 15.
• 3/5: Maximal normal subgroups, commutator subgroup/abelianization, group actions. Reading: Section 15 and 16.
• 3/7: Finish off group actions, Burnside's Lemma. Reading: Section 16, a snippet of 17.
• 3/9: Starting the Rings/Fields section of the course: rings, fields, ring homomorphisms, mention of rings of polynomials, evaluation homomorphisms. Reading: Section 18.
• 3/12: Short review of quotient group computations. Integral domains and zero divisors. Reading: Section 19 (we will skip Section 20, since we have basically covered all of it before). For quotient group computations, see the notes posted on bspace.
• 3/14: Happy Pi day! Finishing integral domains, beginning talking about the quotient field of an integral domain. Reading: Sections 19, 21.
• 3/16: Quotient fields continued. Reading: Section 21.
• 3/19, 3/21, 3/23: This week is all about polynomials. Rings of polynomials, factorization of a polynomial over a field, division theorem for polynomials, factor theorem, irreducible polynomials, irreducibility over Z implies irreducibility over Q; multiplicative group of a finite field. Reading: Sections 22, 23. A couple of the proofs we'll do are omitted from Section 23, but done in Section 45.
• 3/26-3/30: Spring break.
• 4/2: One nice application of Eisenstein's criterion, Euclidean algorithm for F[x] (see Section 46), and unique factorization in F[x]. Possibly starting ideals and quotient rings. Reading: last bit of Section 23, some of Section 46 as applied to polynomial rings. We will not cover Sections 24 or 25.
• 4/4: Midterm 2 in 60 Evans. Covers all material covered in class and homework through 3/23 with a bias towards material covered after 2/22. Midterm 2 solutions. The mean was 35.4/50.
• 4/6: Ideals, ring homomorphisms, and quotient rings. Reading: Section 26.
• 4/9: Fundamental Homomorphism Theorem for rings, prime and maximal ideals. Reading: end of Section 26, Section 27.
• 4/11: More on prime and maximal ideals. Reading: Section 27.
• 4/13: Most of the proof that every PID is a UFD. Absence of unique factorization in Z[sqrt(-5)]. If D is a UFD, then so is D[X] (statement only). Reading: Section 45.
• 4/16: Finishing proof of PID -> UFD, Euclidean domains are PID's, Gaussian integers and multiplicative norms. Reading: Sections 46, 47. Click here for a detailed description of exactly what you are expected to know from Sections 45, 46, and 47.
• 4/18, 4/20: Field extensions, algebraic/transcendental elements, minimal (irreducible) polynomials, vector spaces over a general field. Reading: Sections 29, 30. (Note: I assume you are comfortable with most of the discussion on vector spaces in section 30, so we'll only do a quick review of a lot of it)
• 4/23, 4/25: Algebraic extensions, finite extensions, degree of towers of finite extensions, algebraically closed fields, Fundamental Theorem of Algebra (statement only). Reading: Section 31.
• 4/27: Finishing discussion of algebraic closure, proof that one cannot trisect a 60 degree angle with ruler and compass. Reading: Section 32. NOTE: you won't be tested on section 32 in the final exam.
• 4/30, 5/2, 5/4: RRR week. Review sessions during regular class time. No new material will be covered, of course.
• 5/8: Final Exam 11:30AM-2:30 PM in 3 LeConte. Covers all of the material in the course. Here are the solutions to teh final. The mean was 75.9/100.

• Notes on basic number theory
• M. Hutchings' notes on proofs
• Notes on dihedral groups
• G. Bergman's notes on simplicity of A_n
• Notes on quotient group computations
• G. Bergman's notes on a PID which is not a Euclidean Domain (for your enjoyment, won't be tested on any exam)

• Homework 1, due Friday, 1/27/2012. Some solutions are posted on bspace.
• Homework 2, due Friday, 2/3/2012. Some solutions are posted on bspace.
• Homework 3, due Friday, 2/10/2012. Solutions are posted on bspace.
• Homework 4, due Friday, 2/17/2012. Solutions are posted on bspace.
• Homework 5, due Friday, 2/24/2012. Solutions are posted on bspace.
• No homework due on Friday, 3/2/2012 due to the midterm on 2/29. You might want to look over the practice problems for the midterm instead.
• Homework 6, due Friday, 3/9/2012. Solutions are posted on bspace.
• Homework 7, due Friday, 3/16/2012. Solutions are posted on bspace.
• Homework 8, due Friday, 3/23/2012. Solutions are posted on bspace.
• No homework due on Friday, 4/6/2012 due to the midterm on 4/4. You might want to look over the practice problems for the midterm instead.
• Homework 9, due Friday, 4/13/2012. Solutions are posted on bspace.
• Homework 10, due Friday, 4/20/2012. Solutions are posted on bspace.
• Homework 11, due Friday, 4/27/2012. Solutions are posted on bspace.