The main reference for the course is J. Rotman, "Advanced modern algebra, 3rd edition, part 1" (link only works from ETH computers and via vpn). More information about the course, including further literature, can be found here.
The new exercises will be generally posted here on Fridays. We expect you to look at the problems over the weekend and to prepare questions to ask during the next exercise class on Tuesday/Wednesday.
You are warmly encouraged to hand in your solutions by the following Friday at 12:00 in your assistant's box in HG J 68. Your solutions will be corrected and returned in the following exercise class or put back in the box in HG J 68.
| week | some material covered (with reference from here) | assignment | due by | solutions |
|---|---|---|---|---|
| 15 | Historical introduction on Galois Theory, including Cardano's formula for cubics and Abel-Ruffini's theorem (notes); the Galois group of a field extension (notes; A-5.1,5.2,5.3 and other examples, pages 179-181). | A15 | March, 2nd | S15 |
| 16 | Separable polynomials (A-5.4,5.7 and examples, pages 182-183), extension of field isomorphisms (A-3.98, pages 87-88). Notes. | A16 | March, 9th | S16 |
| 17 | Size of the Galois group (proof of A-5.7, A-5.9, 5.13, pages 183-185), Galois groups of Extensions of finite fields (A-5.13, pages 186-187), A special case in which the Galois group is the whole symmetric group (via Cauchy's theorem), Notes. | A17 | March, 16th | S17 |
| 18 | Transitivity and irreducibility (Prop. A-5.14, page 187), Normal field extensions (Thm. A-5.17, page 190), Solvability by Radicals (see pages 187-188). Notes. | A18 | March, 23th | S18 |
| 19 | Notes. | A19 | April, 6th | S19 |
| 20 | Solvability by radicals, solvable groups, the commutator subgroup, the derived series of a group. Notes. | A20 | April, 13th | S20 |
| 21 | More on solvable groups. Solvability by radicals and solvability of the Galois group. Notes. | A21 | April, 20th | S21 |
| 22 | The subfield fixed by a group of automorphisms. Linear independence of characters. Notes. | A22 | April, 27th | S22 |
| 23 | Characterizations of Galois extensions. Notes. | A23 | May, 4th | S23 |
| 24 | Symmetric functions. The Galois correspondence. Notes and more notes. | A24 | May, 18th | S24 |
| 25 | Simple extensions, the primitive element theorem proved via Galois theory. The norm on a Galois extension. Hilbert 90. Corollary: a degree-p Galois extension of a field containing primitive p-th roots of 1 is pure. Solvability by radicals in characteristic zero. Notes. | A25 | May, 25th | S25 |
| 26 | Cyclotomic polynomials, cyclotomic field extensions. Notes. | A26 | No hand-in | S26 |
All exercise classes are held in English. Please enroll in one exercise group through this link.
| time | room | assistant |
|---|---|---|
| Tue 15-17* | HG E 1.2 | Amanda Jenny |
| Wed 10-12 | CLA E 4 | Pascal Schilde |
| Wed 10-12 | HG E 33.1 | Carlos De la Cruz Mengual |
| Wed 10-12 | HG F 26.5** | Riccardo Ferrario |
| Wed 10-12 | LFW E 13 | Noah Held |
| Wed 10-12 | ML F 40 | Jesse Provost |