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Graduate Program

Information concerning the course Mathematics 552 (Abstract Algebra II) taught in Spring 2017 by J. Tunnell is kept here. The course will be a continuation of Mathematics 551. It will meet Monday and Thursday 12:00-1:20 in Hill 423. The following information is currently available about this course.

 

Text:

Jacobson, "Basic Algebra", Volumes 1 and 2, second edition. These volumes are currently available from Dover (www.doverpublications.com)

Prerequisites:

Any standard course in abstract algebra for undergraduates and/or Math 551. It will be assumed that students understand the concepts of groups, rings, modules, vector space and linear algebra, and finitely generated modules over principal ideal domains.

Description:

Topics: This is the continuation of Math 551, aimed at a discussion of many fundamental algebraic structures. The course will cover the following topics (and perhaps some others).

  1. Basic module theory and introductory homological algebra - most of Chapter 3 and part of Chapter 6 of Basic Algebra II: hom and tensor, projective and injective modules, abelian categories, resolutions, completely reducible modules, the Wedderburn-Artin theorem
  2. Commutative ideal theory and Noetherian rings - part of Chapter 7 of Basic Algebra II: rings of polynomials, localization, primary decomposition theorem, Dedekind domains, Noether normalization
  3. Galois Theory - Chapter 4 of Basic Algebra I and part of Chapter 8 of Basic Algebra II: algebraic and transcendental extensions, separable and normal extensions, the Galois group, solvability of equations by radicals

Course Format: There will be weekly homework assignments, and midterm and final exams.

More Information: Contact J. Tunnell in Hill 546, or email to tunnell@math>

 

Week 1

 Week 2

Week 3

Week 4

Week 5

  • Text Readings: Jacobson II: 4.5, 5.1, 5.2
  • Week 5 Problem set 5 (Due 3/2/17)
  • Week 5 expository reading:
  • Oral Midterm exams will be 3/3, 3/6, 3/7. The possible times are listed here (available 9:00 AM 2/21/17). After you have chosen desired open time slots, email me your choices ranked by preference. I will update the available times as students are scheduled on a first come first served basis.

Week 6

  • Text Readings: Jacobson II: 5.1, 5.2, 5.3
  • Week 6 Problem set 6 (Due 3/9/17)

Week 7

  • Text Readings: Wrapup of Semisimple rings and Representation Theory: Jacobson II: 5.1-5.3, 5.5, 5.6
  • Week 7 Problem set 7 (Due 3/23/17 

Week 8

  • Text Readings: Field extensions, splitting fields, separable polynomials: 
    Jacobson I: 4.1,4.3,4.4
  • Week 8 Problem set 8 (Due 3/30/17)

Week 9

  • Text Readings: Separable polynomials and Galois Theory: Jacobson I: 4.4,4.5
  • Week 9 Problem set 9 (Due 4/6/17)

Week 10

Week 11

  • Text Readings: Solvable groups and Solvability by Radicals: Jacobson I: 4.6,4.7
  • Week 11 Problem set 11 (Due 4/20/17)

Week 12

  • Text Readings: Galois groups of polynomials : Jacobson I: 4.8-4.10,4.13
  • Week 12 Problem set 12 (Due 4/27/17)

Week 13

  • Text Readings: Applications of finite fields; Central Simple Algebras: Jacobson I:4.16, Jacobson II:4.6, 4.7
  • No further weekly problem sets (beyond the previous 12) will be assigned. In lieu of an in-class final exam there will be a take final problem set

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Department of Mathematics

Department of Mathematics
Rutgers University
Hill Center - Busch Campus
110 Frelinghuysen Road
Piscataway, NJ 08854-8019, USA

Phone: +1.848.445.2390
Fax: +1.732.445.5530