|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.
|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.|
|Mon. 10/3||Splitting 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.|