Algebraic Number Theory
Summer 2020/21
Wednesday 12:20 lecture
Thursday 14:00 lecture and exercise (with Giacomo Cherubini) in alternating weeks
all over zoom
Algebraic number theory studies the structure of number fields and forms the basis for most of advanced areas of number theory. In the course we will develop its main tools that are connected to algebraic integers, prime ideals, ideal class group, unit group, and subgroups of the Galois group, including basics of p-adic numbers and applications to Diophantine equations.
Compared to the former lecture NMMB360 this course has more weekly hours (3/1 vs 2/0), and so there will be exercise sessions and we'll cover slightly more material.
There will be an oral exam at the end of the semester.
Credit for the exercises zápočet will be given for solving approx. 3 sets of homework problems.
Office hours
Please email me if you want to discuss anything with me!
Covered material
3. 6. exercise session
27. 5. Frobenius element (Mi, 141-143, 145), Chebotarev density theorem (Mi 8.30-8.34, 8.38, 8.39). video, notes
26. 5. Global fields: extensions of valuations, decomposition groups (Mi, 135-140). video, notes
20. 5. exercise session
19. 5. overview of complete fields, unramified and totally ramified extensions, local fields (Mi 126-131). video, notes
13. 5. Newton's, Hensel's lemma, extensions of non-archimedean absolute values (Mi, pp 118-124). video-a, video-b, video-c, video-d, notes-abc, notes-d
6. 5. exercise session
5. 5. weak approximation theorem, completions (Mi, pages 111-117). video, notes
29. 4. exercise session
28. 4. absolute values, local fields (Mi, Chapter 7, till Theorem 7.12). video, notes
22. 4. FLT (Mi 101-104). video, notes
21. 4. cyclotomic fields (Mi 95-100; proofs of 6.4 and 6.5 will not be at the exam). video, notes
15. 4. exercise session
14. 4. proof of Dirichlet unit theorem. video, notes
8. 4. exercise session
7. 4. proof of Minkowski bound, Dirichlet unit theorem (Mi 85-86). video, notes
1. 4. lattices, proof of Minkowski bound (Mi 73-81). video, notes
31. 3. ideal norm (Mi pages 68-69), Minkowski bound and applications (Mi 70-72). video, notes
25. 3. exercise session
24. 3. ramification (Mi Theorem 3.35). video, notes
18. 3. prime decompositions, efg theorem (IR 12.3, Mi Theorem 3.41). video, notes
17. 3. unique factorization into product of ideals (IR 12.2). video, notes
11. 3. exercise session
10. 3. discriminant (Drápal II.5, see also IR 12.1). existence of integral basis (IR 12.2). video, notes
4. 3. intro, norm and trace, discriminant (Drápal II.5, see also IR 12.1). video, notes
3. 3. recap of quadratic fields from Conrad, and of Pell's equation. video, notes
videos are available here (access info was given at the first lecture - email me if you don't have it)
Exercise problems and videos are here (with the same login and password as the lecture videos)
List of exercise problems that I mentioned during the lectures (compiled by Mikuláš Zindulka)
Recommended reading
[IR] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84 (Second Edition)
[Mi] J. Milne, Algebraic Number Theory, http://www.jmilne.org/math/CourseNotes/ant.html
Poznámky Andrewa Kozlika (pokrývající trochu jiná témata, než co budeme probírat letos)
Skripta z Komutativních okruhů Aleše Drápala
Diplomka Maroše Hrnčiara o řešení diofantických rovnic (a hledání třídových grup)
Moje skripta z Komutativních okruhů
Cílem předmětu je probrat klíčové vlastnosti číselných těles. Budeme předpokládat některé základní pojmy z přednášky Komutativní okruhy, ale téměř nic z teorie čísel.
Základy (zejména jednoznačnou faktorizaci na součin prvoideálů) budou podle kapitoly 12 z [IR], a potom se pustíme do [Mi]: popis prvoideálů (konec kapitoly 3), konečnost třídové grupy (kapitola 4) a struktura grupy jednotek (kapitola 5). Získané výsledky ilustrujeme na příkladech kvadratických a cyklotomických těles a aplikujeme na řešení diofantických rovnic. Nakonec probereme ještě něco ze struktury lokálních těles a Galoisových grup ([Mi], kapitoly 7, 8).
The course page (in Czech) from 2 years ago (this year it will be similar, except for the increased weekly hours).
Spiti valley