Course No.：S070100XJ001
Course Category：basic course of subject
Period/Credits：40/2
Prerequisites：Advianced mathematics, linear algebra, general topology, abstract algebra in college (group theory).
Aims & Requirements：
The course is the basic disciplinary course for all students for the master and PhD degree. It is necessary for the students who want to learn algebraic number theory, algebraic geometry, algebraic topology, computer science and coding theory. The main objects are the students whose majors are not algebra, also the students whose major are algebra can choose it. Through the course students can get the basic training and general knowledge in algebra. The content contains group theory, field theory, Galois theory, ring theory, module theory and so on.
Primary Coverage：
Chapter 1 group theory
Group; homomorphism; concept of representation; simplicity of an alternating group; direct sum and direct product; structure theorem of a finitely generated abelian group; isomorphism theory and decomposition theory; Sylow theory and an introduction to applications of group theory.
Chapter 2 field theory
Prime field; extensions of a field; the unit roots; finite fields; primitive element theorem (simplicity of a finite extension); extension of an infinite field.
Chapter 3 Galois theory
Galois group; normal extension; Galois extension; the fundamental theorem of Galois theory; classical applications of Galois theory.
Chapter 4 ring theory and module theory
Ring; homomorphism and ideal; polynomial ring; Hilbert basis theorem; localization; module; homomorphism; direct sum and direct product; exact sequence; snake lemma.
References：
[1] Thomas. W. Hungerford, Algebra, GTM73
[2] Keqing Feng, Shangzhi Li; Jianguo Cha, <<the introduction to the abstract algebra >>, published by USTC, Beijing, 1988.
Author：Guoping Tang(School of Mathematical Sciences, GUCAS)
Date：June, 2009
