Mathematics 251: Groups, Rings, and Fields (Fall 2002)
Instructor:
Chad Schoen
Description:
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.
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 symmtric 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 (if necessary): symmetric groups, p-groups, Sylow theorems,
solvable groups, group actions, examples. Solvability of polynomial
equations by radicals.
Prerequisites:
Some familiarity with the rudiments of group
theory. Previous exposure to rings would be helpful, but is not essential.
Text:
We will cover the last five chapters in Michael Artin's book,
Algebra, with some deletions and additions.
Course Website:
For more information see http://www.math.duke.edu/faculty/schoen/alg2002home.html
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Last modified: 27 March 2002