Modern Algebra I

Detailed Information

Course Description

The ultimate goal of this course is to introduce the main theorems and standard examples in Galois theory. Along the way we will cover the fundamentals of groups, commutative rings, and fields.

The roots of Galois theory lead back to problems posed by the ancient Greeks and their predecessors. Greek geometers achieved remarkable constructions with ruler and compass, but a number of simple, nagging, problems remained unresolved until the Rennaisance. For example: Is it possible to trisect an arbitrary angle? Which regular polygons are constructible? In algebra, several civilizations investigated the solution of a quadratic equation (see MathWorld article ). The attempt to find a solution for higher degree equations was another project that occupied numerous mathematicians. The resolution of these ancient questions culminated in Galois' theory of fields. It is a delightful subject, and the modern treatment highlights the interplay between three key areas of algebra: groups, rings and fields.

Texts

This book is a concise and direct treatment of the fundamentals of graduate level algebra.

This has been the standard text for undergraduate algebra at SDSU for a few years. It will be useful for review of material that is covered tersely in Ash's book.

Prerequisites

• The integers (H 1.1-3, 2.1-2): The division theorem, greatest common divisor, Euclidean algorithm, prime numbers, unique factorization, modular arithmetic.
• Polynomial rings in one variable over a field (e.g. the rationals) (H. 4.1-4, 5.1-2): The division theorem, greatest common divisor, Euclidean algorithm, irreducible polynomials, the correspondence between linear factors of a polynomial and roots, unique factorization, congruence modulo a polynomial.
• Commutative rings (H 3.1-3, 6.1): We will only treat commutative rings, and they will usually be derived from the integers or a polynomial ring over the rationals. You should be familiar with the basic definitions: ideals, homomorphisms, integral domain.
• Linear Algebra: Vector space over a field, nullspace, subspace, dimension, linear independence, spanning set, basis.
• Groups (H. 7.1-5): Definitions of group, subgroup, homomorphism. The most basic results (e.g. uniqueness of the inverse). Standard examples like Z_n (integers mod n), D_n (the dihedral group), S_n (the symmetric group). Cosets, index, Lagrange's theorem.

Foundational Topics

Some of the following topics will be familiar from your undergraduate course. We will cover them in greater depth, and with more attention to details. I expect this material to take 6-9 weeks.

• Groups: Normal subgroups and the isomorphism theorems. Groups acting on sets, orbits and stabilizers. The Sylow theorems.
• Commutative Rings: Ideals, homomorphisms and quotient rings. The isomorphims theorems. Prime and maximal ideals. Criteria for irreducibility of polynomials.
• Fields: Field extensions. Adjoining a root of a polynomial. Splitting field of a polynomial.

Galois Theory

• Separable, inseparable, and normal field extensions.
• The Galois group of a field extension.
• The fundamental theorem: relating intermediate fields of K/F to subgroups of the Galois group of K/F
• Standard examples: finite fields, cyclotomic fields, solvability by radicals, constructable numbers.

Grading

There will be several (6-8) homework assignments a midterm and a final exam. The final grade will be weighted as follows.

Problem Sets	35%
Midterm	35%
Final Exam	30%