Abstract Algebra --- Rings and Fields
| General Information | Syllabus | HW Assignments |
|---|
Syllabus
The following is a tentative syllabus for the course. This page will be updated irregularly.On the other hand, the weekly syllabus contained in the Homework Assignments page will always be accurate.
| Lectures | Sections in Text | Brief Description |
|---|---|---|
| Week 1 | II.1, II.3, IV.1(part) | A quick review of rings
(examples, properties, homomorphism theorems), polynomial rings in several variables, division algorithm over commutative rings with 1, polynomials versus polynomial maps, prime and maximal ideals. |
| Week 2 | II.2, II.5, IV.1, IV.2 | operations on ideals,
correspondence theorem, CRT, irreducibles and prime elements, UFDs, PIDs, Noetherian rings, Euclidean domains |
| Week 3 | IV.2, IV.3, IV.4, random tidbits | Gauss's
lemma and corollaries, Irreducibility tests, Hilbert's Basis
Theorem, Cyclotomic polynomials, start finite and algebraic field extensions |
| Week 4 | V.1, V.2 | Finite, and algebraic extensions, splitting fields, composites and distinguished classes of extensions |
| Week 5 | IV.1, V.2, supplementary material | tests for separability, irreducibility of cylotomic polynomials, finite fields, extending embeddings, existence and uniqueness of splitting fields, algebraic closures and uniqueness |
| Week 6 | V.2, V.3, V.4, supplementary | Compass and straightedge constructions, embeddings, normality for general algebraic extensions, separability, begin Galois theory |
| Week 7 | V.4, VI.1 | Equivalent versions of the FTGT for finite extensions, examples: cyclotomic, biquadratic, $x^3−2$, $x^4−2$ over $\mathbb Q$, normality in Galois extensions, composites and liftings of Galois extensions |
| Week 8 | VI.1, VI.2,VI.4, VI.6 | Finite fields, irreducibles over $\mathbb F_p$, finite abelian groups are Galois groups, prime cyclotomic fields and primitive elements, Artin's theorem on characters, norm and trace, solvability by radicals, Hilbert's Theorem 90 |
| Week 9 | VI.2 - VI.7, class notes | Solvability by radicals, Insolvability of the quintic, The general polynomial of degree $n$. |
T. R. Shemanske
Last updated July 15, 2022 11:51:41 EDT