14.15-16.00 room B322
I. Stewart. (2023) GALOIS THEORY, fifth ed.
D. Cox, J. Little, D. O’Shea. (2015). IDEALS, VARIETIES, AND ALGORITHMS, fourth ed.
| week | lecture (Tu) | exercise (Wed) | lecture (Thu) |
| 3 | 16.01. field extensions, field extensions as vector spaces, the Tower Law |
17.01. | 18.01. algebraic vs. transcendental, minimal polynomial, classification of simple algebraic and simple transcendental extensions |
| 4 | 23.01. primitive element theorem, the subfield of algebraic numbers, algebraic numbers as eigenvalues |
24.01. | 25.01. polynomials for algebraic sums and products, constructible numbers and quadratic field extensions |
| 5 | 30.01. impossible constructions, origami numbers |
31.01. | 01.02. multivariate polynomial ideals and varieties |
| 6 | 06.02. monomial orders, multivariate polynomial division |
07.02. | 08.02. monomial ideals, Dickson's Lemma, ideal of leading terms |
| 7 | 13.02. Hilbert's Basis Theorem, Gröbner bases, Ascending Chain Condition, variety of an ideal, S-polynomials |
14.02. | 15.02. Buchberger's criterion, Buchberger's algorithm, reduced Gröbner bases |
| 8 | 20.02. Sage sample coprimality as a Buchberger shortcut, polynomial computations in SageMath |
21.02. | 22.02. standard representation, lcm representation, Elimination Theorem, Extension Theorem |
| 9 | 27.02. proof of the Extension Theorem, Geometric Extension Theorem |
28.02. | 29.02. polynomial and rational implicitization |
| 10 | - | - | - |
| 11 | 12.03. Weak Nullstellensatz, Hilbert's Nullstellensatz, radical ideals, Strong Nullstellensatz |
13.03. | 14.03. ideal-variety correspondence, radical membership, square-free polynomials |
| 12 | 19.03. sums products and intersections of ideals, lcm |
20.03. | 21.03. computing intersection ideals, Zariski closure |
| 13 | 26.03. ideal quotients, ideal saturations |
27.03. | - |
| 14 | - | - | 04.04. computing ideal quotients and saturations, irreducible varieties, prime ideals, maximal ideals |
| 15 | 09.04. maximal ideals are points, irreducible decompositions, minimal decompositions |
10.04. | 11.04. the closure theorem |
| 16 | 16.04. polynomial mappings on varieties, coordinate ring, pullbacks, isomorphic varieties |
17.04. | 18.04. quotients by ideals as vector spaces, finiteness theorem |
| 17 | 23.04. zero dimensional ideals, ideal-variety correspondence for subvarieties, function field of a variety |
24.04. | 25.04. rational mappings, birational equivalence |
| 18 | 30.04. recap |
- | 02.05. exercise |