Math 113: Introduction to Abstract Algebra—Summer 2015

Time and place
Course content and prerequisites
Exams, homework and grading policy
Syllabus, reading and homework assignments


(8/7) Please take a few minutes to give your Course Evaluation for this class.
(8/6) Along with PS 8, I have posted an outline of topics and review exercises for the final exam.
(8/6) Correction on PS 7: 9.2 3 should be 9.2 2, as you probably realized from the hint.
(7/27) Correction on PS 6: omit problem 6.8.5(c).
(6/25) We now have a a bCourses page. You can log in with your CalNet ID to view your scores and homework solutions.
(6/23) Correction: the final exam is on August 13, the last day of class, not Aug 14 as I previously posted by mistake.
(5/13) Welcome to Math 113! Check here for updates.


Mark Haiman,
Office hours Monday, Tuesday and Wednesday 12-1, or by appointment.

Time and place

Monday through Thursday 10-12, Room 289 Cory Hall.
Each meeting will consist of a mix of lectures, discussion and in-class problem-solving activities.

Course content and prerequisites

This course covers three main topics: (1) the theory of groups, with applications to symmetry; (2) commutative rings, focusing on the ring of integers and polynomial rings; and (3) fields, field extensions, and Galois theory, with applications to geometry and algebra.

The official prerequisite for this course is Math 54 or equivalent preparation in linear algebra. It is also helpful to have previous experience with logic and proofs, sets, and mathematical induction, for instance as covered in Math 55.

You will find some review material on logic, sets, induction, complex numbers and linear algebra in Appendices A-E of the course textbook.


Frederick M. Goodman, Algebra: Abstract and Concrete, Edition 2.6.
Follow this link to download the text in PDF format, free of charge. The author requests that you make a donation to a charitable organization of your choice in lieu of a royalty.

Exams, homework and grading policy

Grades will be based on two midterm exams, a final exam, and weekly homework assignments, according to the formula: Homework 15%, Midterms 25% each, Final 35%.

Exam dates:

All exams take place during the usual two-hour class period. You will need to bring your own `blue books' and scratch paper. The final exam will cover all course material but with extra emphasis on topics not yet covered on midterm exams.

Exam policies: you may consult handwritten notes which you prepare in advance. No other books, printed materials, or electronic devices may be used.

In general there will be no make-up exams and no dropped scores. In case of illness or emergency, I will consider making allowances for missed exams on an individual basis, provided you have a valid reason and contact me as soon as possible.

Homework policies: usually two or three problems from each homework set will be chosen for grading.

You may work with others to solve homework problems, but you must write your solutions indiviually. Copying solutions from other students, classes in previous years, the internet, or elsewhere is not allowed.

You can view your homework and exam scores on the class bCourses page (choose "Grades" from the bCourses menu). I will also use bCourses to post homework solutions.

Syllabus, reading and homework assignments

All homework is due on the Thursday of the week for which it is assigned.

Chapter numbers in reading assignments refer to Goodman. I also plan to distribute supplementary notes on Euler's function φ(n), geometric constructions, finite fields, and the Galois group of a general polyomial.

Week Topics Reading Homework
June 22-25 Groups of symmetries and permutations. Divisibility, GCD, prime factorization. Modular arithmetic. 1.1-1.7 Problem Set 1, Solutions*
June 29-July 2 Chinese remainder theorem. Euler's function φ(n). Groups, subgroups and cyclic groups. Dihedral groups. Homomorphisms and normal subgroups. Cosets, Lagrange's theorem, order and index of subgroups. Quotient groups. 1.10, 2.1-2.5, 2.7, Notes on φ(n) Problem Set 2, Solutions
July 6-9 Subgroups of G/N and Zn. Homomorphism theorems. Partitions and equivalence relations. Direct products. 2.6-2.7, 3.1 Problem Set 3, Solutions.

Midterm 1, Solutions
July 13-16 Semidirect products. Smith normal form and structure of finitely generated abelian groups. Symmetries of regular polyhera. Group actions, orbits, stabilizers; Burnside's lemma. Automorphisms, class equation, solvability of p-groups. 3.2, 3.5-3.6, 4.1-4.3, 5.1-5.4 (through Cor. 5.4.5), 11.1, 11.4 Problem Set 4, Solutions
July 20-23 Polynomials. Rings and fields. Homomorphisms, ideals, direct sums, quotients. Integral domains, relationship between divisibility and principal ideals, PID's. Examples: fields (familiar and new), polynomial rings, the ring of integers. 1.8, 1.11, 6.1-6.4 Problem Set 5, Solutions
July 27-30 PID implies UFD. Prime and maximal ideals. Construction of fields K[x]/(p(x)) where p(x) is an irreducible polynomial. Fraction field of an integral domain. Divisibility and gcd in a UFD. Gauss's Lemma. R UFD implies R[x] UFD. Irreducibility tests. 6.5, 6.6 through page 313, 6.8, 7.1-7.2 Problem Set 6**, Solutions.

Midterm 2, Solutions
Aug 3-6 Vector spaces over a field. Field extensions, dimension, algebraic and transcendental elements. Finite and algebraic extensions. Construction of extension K(α) of K. Roots and splitting fields. Impossibility of angle trisection. Finite fields. Preview of Galois theory for splitting fields in the complex numbers. Guide to field theory; parts of 3.3, 7.3-7.5, 9.1-9.2 Problem Set 7**, Solutions
Aug 10-13 Separability criteria. Definition and characterization of Galois extensions. Main theorem of Galois theory. Galois group of a general polynomial. Solvable groups, simple groups, unsolvability of the general quintic by radicals. Notes on finite fields; parts of 9.3-9.7, 10.1-10.6

Review Guide
Problem Set 8, Solutions

Final Exam, Solutions

Online Course Evaluation

*Homework problems chosen for grading:

**Corrections: PS 6—omit 6.8.5(c); PS 7—9.2 3 should be 9.2 2.

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