Course plan
On this page, there will be a course plan that will be adapted and extended during time. Here are typed lecture notes of some students from last year's course Typed notes. The content will be quite similar to this year's syllabus; note that there are some typos in these notes.
| Day | Book chapters | Topic covered | |||
|---|---|---|---|---|---|
| Thursday 09.01. | Chapters 11.1,11.2 | Historical Background and motivation, Definitions of integral domains, divisibility, irreducibles, primes; Definition of UFDs and proving the equivalence of primes and irreducibles in UFDs | |||
| Friday 10.01. | Chapters 11.2,11.3 | PIDs, proof of PID → UFD, definitions of GCD, Euclidean algorithm | |||
| Thursday 16.01. | Chapters 11.3,11.4,15.1 | Euclidean domains, polynomial rings, irreducibility in Z and Q (Gauss' Lemma) | |||
| Friday 17.01. | Chapter 15.1 | Eisenstein criterion, Properties of polynomial rings over fields, field extensions | |||
| Thursday 23.01. | - | Exercise class, find the exercise sheet here | |||
| Friday 24.01. | Chapters 15.2, 15.3 | Adjunction of roots, Kronecker Theorem, algebraic and transcendental extensions | |||
| Thursday 30.01 | Chapter 15.3, 15.4 | Algebraic and transcendental extensions, algebraic closures, algebraically closed fields | |||
| Friday 31.01. | Chapter 15.4 | Proof of the unique algebraic closure | |||
| Thursday 6.02. | Chapter 16.1, 16.2 | Splitting fields and normal extensions | |||
| Friday 7.02. | Chapter 16.3,16.4 | ||||
| Thursday 13.02. | - | Exercise class, find the exercise sheet here | |||
| Friday 14.02. | Chapter 16.4,16.5 | Properties of finite fields, multiple roots, definition of separable and perfect | |||
| Thursday 20.02. | Chapter 17.1 | Galois groups and fixed fields | |||
| Friday 21.02. | Chapter 17.1,17.2 | Galois groups and fixed fields, Fundamental Theorem of Galois theory, part I | |||
| Thursday 27.02. | Chapter 17.2 | Fundamental Theorem of Galois theory, part II | |||
| Friday 28.02. | Chapter 17.3,18.1 | Fundamental Theorem of Algebra, roots of unity | |||
| Thursday 06.03. | - | Exercise class, find the exercise sheet here. | |||
| Friday 07.03. | Chapter 18.1 | Cyclotomic polynomials | |||
| Thursday 13.03. | Chapter 18.2 | Cyclic extensions | |||
| Friday 14.03. | Chapter 18.3 | Radical extensions | |||
| Thursday 20.03. | Chapter 6.1,6.2 | Excursion to Group theory: Solvable groups | |||
| Friday 21.03. | Chapter 18.3 | Proof of Theorem 3.2 | |||
| Thursday 27.03. | - | Exercise class, find the exercise sheet here. | |||
| Friday 28.03. | Chapter 4.5, 6.1 | Excursion to symmetric Groups: Unsolvability of S_n for n > 4 | |||
| Thursday 03.04 | Chapter 18.3,18.5 | Unsolvable polynomials by radicals, Construction with ruler and compass (constructible points form a field) | |||
| Friday 04.04 | Chapter 18.5 | Construction with ruler and compass (characterization of constructible points, first applications) | |||
| Thursday 10.04 | Chapter 18.5 | Construction with ruler and compass (construction of angles, construction of regular n-gons) | |||
| Friday 11.04 | - | Exercise class, find the exercise sheet here. |
There are exercise classes as soon as we have enough material to discuss. All exercises are voluntary! You will find exercise sheets here about a week in advance to the exercise class. Students should present their solutions on the blackboard to train presentation (and English) skills. The solutions won't be graded but I hope there will be enough volunteers, otherwise I have to replace these sessions with classical lectures. It is not only allowed but highly appreciated that the exercises are discussed and solved in small groups. The most active students will receive some goodies at the end of the lecture!