Math 251: Introductory Graduate Algebra
Instructor: Chad Schoen
Course Goals:
This one semester course provides a systematic introduction to algebra with emphasis
on Galois theory. It is primarily intended for students who plan to
become professional mathematicians. It is customarily taken by first year
graduate students in mathematics. It is a prerequisite for all more
advanced courses in algebra and algebraic geometry.
Prerequisites:
Some familiarity with the rudiments of group theory. Previous exposure to
rings would be helpful, but is not essential.
Topics:
Review of rings: ideals, principal ideal domains,
Chinese Remainder Theorem for comaximal ideals, localization.
Modules over a Principal Ideal Domain with applications
to classification of finitely generated Abelian groups and Jordan canonical
form for matrices. Gauss' Lemma, irreducibility criteria for polynomials.
Abstract construction of algebraic field extensions, splitting fields.
Classification of finite fields. Quadratic, biquadratic, and cubic extensions,
fundamental theorem on symmetric functions, primitive element theorem,
Galois extensions and separable splitting fields, the subgroup-intermediate
field correspondence of Galois theory, Kummer theory, cyclotomic extensions.
Rapid review of group theory (solvable groups, symmetric groups, simplicity of
the alternating group on 5 letters). Solvability of
polynomial equations by radicals.
Text:
We will cover the last five chapters in Michael Artin's
book, Algebra, with some deletions and additions.
Grading
Lengthy homework assignments are given every week.
These are then written up, turned in and graded.
A final exam is also expected.