Basic algebraic number theory 2024/25
Summer 2024/25
Monday 12:20 lectures in K2
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 course NMAG430 this course has fewer weekly hours (2/0 vs 3/1), and so there will be no formal exercise sessions and we’ll cover less material at a more leisurable pace. There will be a new continuation advanced course in the Fall.
There will be an oral exam at the end of the semester.
I plan to teach the course in English and to prepare lecture notes during the semester.
Office hours
Please email me if you want to discuss anything with me!
Covered material
future program is very preliminary
19. 5. buffer
12. 5. FLT
5. 5. cyclotomic fields
28. 4. proof of Dirichlet unit theorem
21. 4. Easter
21. 4. Easter
14. 4. proof of Minkowski bound, Dirichlet unit theorem
7. 4. lattices, proof of Minkowski bound
31. 3. ideal norm, Minkowski bound and applications
24. 3. ramification
17. 3. prime decompositions, efg theorem
10. 3. unique factorization into product of ideals
3. 3. existence of integral basis
24. 2. norm and trace, discriminant
17. 2. introduction
24. 3. ramification
17. 3. prime decompositions, efg theorem
10. 3. unique factorization into product of ideals
3. 3. existence of integral basis
24. 2. norm and trace, discriminant
17. 2. introduction
Recommended reading
Main sources
[IR] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84 (Second Edition)
[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
Other sources
[La] Serge Lang, Algebraic Number Theory, GTM 110, 1994.
[BS] E.I. Borevič, I.R. Šafarevič: Number Theory, Academic Press 1966.
[Co] H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin 1996.
[FT] A. Frőhlich, M.J. Taylor, Algebraic number theory, Cambridge University Press, Cambridge 1991.
In Czech
My lecture notes for Introduction to commutative algebra
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)
The course page (in Czech) from 6 years ago (this year it will be somewhat similar). 4 years ago the course had more weekly hours, and so we covered more material.
