| Day | Topics | |
|---|---|---|
| Wed. 8/31 | Introduction, some simple fields.
The classical problems: solution of equations, ruler and compass constructions. |
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| Wed. 9/7 | Rings and fields: numerous examples. Ring homomorphisms. |
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| Mon. 9/12 | Rings of fractions. Ideals and quotient rings. |
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| 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. |
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| Mon. 9/26 | Field extensions, and roots of polynomials. | |
| Wed. 9/28 | Derivatives. | |
| Mon. 10/3 | Splitting fields. Finite fields. |
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| Wed. 10/5 | Structure of finite fields: primitive elements, subfield structure. |
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| Mon. 10/10 | The Frobenius map. Separable and inseparable field extensions. An inseparable field extension. |
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| Wed. 10/12 | Uniqueness of splitting field. The polynomial ring over a UFD is a UFD. |
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| 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. |
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| 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. |