Difference between revisions of "Math 372: Abstract Algebra 2."
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Revision as of 16:30, 19 January 2011
Contents
Catalog Information
Title
Abstract Algebra 2.
(Credit Hours:Lecture Hours:Lab Hours)
(3:3:0)
Offered
F, W
Prerequisite
Description
Fields, Galois theory, solvability of polynomials by radicals.
Desired Learning Outcomes
Prerequisites
Students need to have mastered Math 313 (linear algebra) and Math 371 (group theory and ring theory) prior to enrolling in Math 372.
This is a second course in abstract algebra focusing on field theory. The course is aimed at undergraduate mathematics majors, and it is strongly recommended for students intending to complete a graduate degree in mathematics. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and nonexamples of the various concepts. The students should be able to demonstrate their mastery by solving nontrivial problems related to these concepts, and by proving simple (but nontrivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.
 Ring Theory
 Ideals and ring homomorphisms
 Quotient rings
 Prime and maximal ideals
 Polynomial rings over fields
 Factorization in polynomial rings
 Irreducible polynomials
 Polynomial division algorithm
 Field Theory
 Extensions of fields
 Field extensions via quotients in polynomial rings
 Automorphisms of fields
 Finite fields
 Fields of characteristic 0 and prime characteristic
 Splitting fields
 Galois extensions and Galois groups
 The Galois correspondence
 Independence of characters
 Fundamental Theorem of Galois Theory
 Fundamental Theorem of Algebra
 Roots of unity
 Solvability by radicals
 Ruler and compass constructions
 Insolvability of the quintic
Textbooks
Possible textbooks for this course include (but are not limited to):
 Joseph Rotman, Galois Theory (Second Edition), Springer, 1998.
 David Dummit and Richard Foote, Abstract Algebra (Third Edition), Wiley, 2003. (The chapters on fields and Galois theory, and some of the material on rings.)
 Ian Stewart, Galois Theory (Third Edition), Chapman Hall, 2004.
Additional topics
The instructor may cover additional topics beyond the minimal requirements. Possible topics include (but are not limited to): introduction to algebraic numbers, applications of field extensions to cryptography, applications of field extensions to diophantine analysis, relations of field theory to algebraic geometry, construction of algebraically closed fields using Zorn's lemma, introduction to computer calculations in abstract algebra.