Modern Algebra II

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 fundamental theory of groups, 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.

We will start with a review of the group axioms, basic properties, subgroups and homomorphisms by working with lots of examples. We will do the same for rings and fields, with a focus on constructing new fields by computing modulo an irreducible polynomial. The first part of the course ends with an exploration of the connection between solving polynomials and field extensions.

The second part of the course will develop the key results of group theory: quotient groups, the isomorphism theorems, the Sylow theorems, group actions, and solvability of a group. The third part of the course will start with the study of field extensions, culminating with Galois's theorem connecting field extensions to automorphism groups. We will then apply the theory to as many examples and classical problems as time allows.

Resources

• Ash, Robert, Abstract Algebra: The basic graduate year . available online

  This book is a concise and direct treatment of the fundamentals of graduate level algebra. Since it is a bit too concise in some areas I will write some lecture notes. I also suggest having a copy of a good undergraduate level text to have access to more detailed explanations and lots of examples. Three such examples are below.

• Hungerford, Abstract Algebra: An Introduction 2nd ed.

  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.

• Gallian, Contemporary Abstract Algebra.
• Judson and Beezer, Abstract Algebra: Theory and Applications

  This is another good undergraduate algebra text. Available free online. See the button for Sage and AATA. Download "Sage Worksheet Collection." Open the zip file from the sage notebook.

• SDSU Sage Tutorial Updated 1/28/12. Expect updates throughout the semester.

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.
• 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).

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, constructible 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	30%
Midterm	30%
Final Exam	40%