The course is mainly devoted to the classical algebraic structures considered from the modern point of view. The categorical language is developed in the amount sufficient for considering the categories of groups, rings, modules, field extensions, etc., as well as for discussing the functors relating to them. The problem of classification up to isomorphism is going to be the central one. Special attention will be paid to the structures related to classical arithmetic.
Prerequisites
Abstract Algebra I and Linear Algebra (Advanced Linear Algebra from the list of our courses).
Curriculum:
- Basic group theory. Cosets, quotients, normal subgroups.
- Existence of elements of prime order.
- Actions of finite groups on sets. Orbits, stabilizers.
- Actions of \(p\)-groups on finite sets and the number of fixed points.
- Sylow’s theorems. Existence, conjugacy, number of Sylow subgroups.
- Simple groups. Solvable groups. Nilpotent groups.
- Algebraic extensions of fields.
- Separable field extensions. Normal extensions.
- Galois extensions. Galois correspondence.
- Solvability of polynomials in radicals.
- Complements to the course on basic representation theory.
- Irreducible representations. Schur’s lemma.
- Semisimple algebras and modules.
- Group algebra. Maschke’s theorem.
- Application of the theory of characters to the structure of finite groups. Solvability of groups of order \(p^n q^m\).
Textbooks
- E. B. Vinberg, A Course in Algebra, No. 56, American Mathematical Society, 2003.
- K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Vol. 84, Springer Science & Business Media, 2013.
- S. Lang, Algebra, revised third edition, Graduate Texts in Mathematics 211, Springer, 2002.
- D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004.
- R. B. Ash, Basic Abstract Algebra: For Graduate Students and Advanced Undergraduates, Courier Corporation, 2013.