| Lectures | Sections in Text | Brief Description |
|---|---|---|
| Week 1 |
[L] II.1 - II.2 (II.3); IV.1 [J] Chapters 16-17 |
Overview of the spectrum: integral domains, UFDs, Noetherian domains, PIDs, Euclidean domains. Characteristic of a ring, prime subfield, polynomials in one and several variables and their universal mapping properites, general division algorithm, roots of polynomials, prime and maximal ideals, operations on ideals. |
| Week 2 | [L] II.5, IV.1-3 | Correspondence theorem, CRT, irreducible and prime elements, UFDs, PIDs, Noetherian rings, Gauss's lemma and corollaries, irreducibility tests? |
| Week 3 | [L] IV.3, IV.4, V.1 | Irreducibility tests, cyclotomic polynomials, begin field theory |
| Week 4 | [L] V.1, V.2 | Finite and algebraic extensions, splitting fields, composite fields, distinguished classes of extensions |
| Week 5 | [L] IV.1, V.1, V.2 | Irreducibility of cyclotomic polynomials, extending embeddings, uniqueness of splitting fields, separability, finite fields, algebraic closure and algebraically closed fields. |
| Week 6 | [L] V.2, V.3, V.4 | Construction and uniqueness of algebraic closures (summarized), separability, separable degree, primitive element theorem, being Galois theory. |
| Week 7 | [L] VI.1, supplements | Galois correspondence, equivalent versions of correspondence for finite extensions, examples. |
| Week 8 | [L] VI.1-VI.7 | More examples, Normality of intermediate extensions, Galois extensions under lifts and composites, counting irreducibles in $\mathbb F_p$, Artin's theorem on characters. |
| Week 9 | [L] VI.2-VI.7 | Norm and Trace, Hilbert's Theorem 90, Cyclic and radical extensions, solvability by radicals |
| 3/12 | Final Exam | 8-11am |