# Algebraic Structures I

## Fall 2017, Duke University

General information | Course description | Lecture notes | Assignments | Homework schedule | Grading | Links | Fine print

## General information

Lectures: Wednesday and Friday, 15:05 – 16:20, Physics Building 227

Text: Abstract Algebra, by David S. Dummit and Richard M. Foote (third edition)

Secondary texts:

• Artin: Algebra
• Lang: Algebra
• Herstein: Topics in Algebra

### Contact information for the Instructor

Name: Professor Ezra Miller
Address: Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320
Office: Physics 209
Phone: (919) 660-2846
Email: ezramath.duke.edu
Webpage: https://math.duke.edu/people/ezra-miller
Course webpage: you're already looking at it... but it's https://services.math.duke.edu/~ezra/501/501.html
Office hours: Tuesday 14:00 – 15:00 & Wednesday 12:00 – 13:15, in Physics 209

## Course description

Course content:

• groups
• subgroups
• homomorphisms
• cosets
• quotient groups
• symmetry
• group actions
• permutation representations
• Sylow theorems
• rings
• ideals
• polynomials
• fractions
• modules
• structure theorem over PID
Groups encapsulate the notion of symmetry. They constitute the simplest way to compose a single type of invertible operation, such as addition of numbers, multiplication of matrices with nonzero determinant, rotations of spheres, rigid motions of polygons and polyhedra, or permutations of sets of objects. The study of groups in this course includes decompositions, enumerations, quotients, and actions.

Rings combine two operations: addition and multiplication. In familiar situations, particularly the integers and univariate polynomials, interactions between the two operations lead to fundamental theorems concerning factorization into primes. What results is a main goal of the course: a single structure theorem that classifies all finite abelian groups and also produces Jordan canonical forms of linear transformations.

Prerequisites: Math 501 is a demanding course. Students are expected to have a firm grasp of linear algebra before beginning the course. In addition, it is expected that every student begins the course comfortable and proficient at writing rigorous mathematical arguments (proofs).

## Assignments

• Reading assignments are included at the top of each homework and midterm.
• Due dates for the five homework assignments this semester are listed in the table below.
• All assignments, including the midterms, will be take-home. All are due at the start of class time on the due date.
• All solutions you turn in, including midterms, homework, and term projects, must be typewritten using the provided LaTeX template. Communicating your ideas is an integral part of mathematics. In addition to the usual PDF files, LaTeX source files for each of the homework assignments as well as each of the midterms will be provided. You should use these as LaTeX templates for your solutions, by filling in your responses in those files. I am happy to answer any questions you might have about LaTeX, although you should ask your classmates first.
• Turn in your homework solutions to me by email. Send at least your .tex file (the grader or I may comment on your TeX usage); if you include your .pdf file as well then it can serve as verification that your system produces the same output as ours do. Do not email your midterm solutions to the grader; email them only to me.
• Collaboration on homework is encouraged, as long as each person understands the solutions, writes them up using their own words, and indicates—on the homework page—who their collaborators were.
• In contrast, no collaboration or consultation of human or electronic sources—except for the "Text" listed above"—is allowed for either of the two midterms or the final exam.
• You must cite sources in your solutions. If you rely on so-and-so's theorem, then you must state the theorem and tell me where you found it. Be specific: "the dual rank theorem" is not precise; in contrast, "[Climenhaga, Theorem 5.10]" is. Theorems are often known by many names, so I'm likely not to recognize many theorems by names you might attach.
Check here two weeks before each homework is due, or one week before each exam is due, for the specifics of the assignments. If an assignment hasn't been posted, and you think it should have been, please do email me. Sometimes I encounter problems (such as, for example, the department's servers going down) while posting assignments; other times, I might simply have neglected to copy the assignment into the appropriate directory, or to set the permissions properly. (I do try to check these things, of course, but sometimes web pages act differently for users inside and outside the Math department.)

Homework #1 Wed. 13 September in PDF or LaTeX §1.1–1.6, §2.1, §2.3
Homework #2 Wed. 27 September in PDF or LaTeX §2.2, §2.4, §3.1–3.3, §5.1
Midterm 1 Wed.   4 October in PDF or LaTeX §1.7, §4.1, §3.5
Homework #3    Fri. 20 October in PDF or LaTeX §4.2–4.6, §5.3–5.5, §6.3
Homework #4    Fri.   3 November in PDF or LaTeX §7.1–7.4, §7.6
Midterm 2    Fri. 10 November in PDF or LaTeX §8.1–8.3, §7.5
Homework #5 Wed.   6 December in PDF or LaTeX §9.1–9.2, §10.1–10.3, §10.5, §12.1, §15.4
Final exam Wed. 13 December, noon in PDF or LaTeX

• 50% Homework
• 30% Midterms
• 20% Final exam
Participation in class discussion and office hours can contribute substantially to your homework score.

### The fine print

I will do my best to keep this web page for Math 403 current, but this web page is not intended to be a substitute for attendance. Students are held responsible for all announcements and all course content delivered in class.
Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage many years ago.

The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.

ezramath.duke.edu
Fri Dec 1 02:20:57 EST 2017

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