Modern Algebra I: Math 627A
Fall 2005 Schedule


A best approximation.

Day Topics
Wed. 8/31 Introduction, some simple fields.
The classical problems:
   solution of equations,
   ruler and compass constructions.
Wed. 9/7 Rings and fields: numerous examples.
Ring homomorphisms.
Mon. 9/12 Rings of fractions.
Ideals and quotient rings.
Wed. 9/14 More on ideals.
Mon. 9/19 Polynomial ring over a field.
Wed. 9/21 Maximal and prime ideals
Quotients of a polynomial ring over a field.
Mon. 9/26 Field extensions, and roots of polynomials.
Wed. 9/28 Derivatives.
Mon. 10/3 Splitting fields.
Finite fields.
Wed. 10/5 Structure of finite fields:
primitive elements, subfield structure.
Mon. 10/10 The Frobenius map.
Separable and inseparable field extensions.
An inseparable field extension.
Wed. 10/12 Uniqueness of splitting field.
The polynomial ring over a UFD is a UFD.
Mon. 10/17 Criteria for irreduciblity.
Wed. 10/19 Solving equations.
Mon. 10/24 The Galois group of a field extension.
Wed. 10/26 Groups, subgroups.
Mon. 10/31 Normal subgroups. Group homomorphisms.
Wed. 11/2 Isomorphism theorems.
Properties of normal subgroups.
Free group, direct sum of groups.
Mon. 11/7 Abelian groups.
Wed. 11/9 Classification of Abelian groups.
Mon. 11/14 Solvable groups.
Wed. 11/16 Sylow theorem.
Field extensions by roots of unity.
Mon. 11/21 Splitting field of x^n-c.
Solvability by radicals.
Mon. 11/28 Solvability by radicals
Wed. 11/30 Constructible numbers
Mon. 12/5 Constructibility of regular polygons.
Normal and Galois extensions.
Wed. 12/7 Independence of characters.
The Fundamental Theorem of Galois theory
Mon. 12/12 The Fundamental Theorem. Examples.