Basic algebraic number theory 2025/26
Summer 2025/26
Monday 9:00 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.
Thematically, the course is a direct continuation of Chapter 4 of NMAG305 (taught in Czech), although strictly speaking, the only prerequisites come from field theory (mostly Chapter 2). The course is suitable not only for Master’s students, but also for interested 3rd year Bachelor’s students. In the Fall, there is an advanced continuation course NMAL431.
There will be an oral exam.
I plan to teach the course in English and to prepare lecture notes during the semester. Here’s a rough version of the notes – I suggest you print it maybe only 1-2 weeks ahead of class, probably there’ll be quite a lot of changes.
Office hours
Please email me if you want to discuss anything with me!
Covered material
very preliminary
18. 5. buffer
11. 5. cyclotomic fields and FLT
4. 5. cyclotomic fields
27. 4. proof of Dirichlet unit theorem
20. 4. proof of Minkowski bound, Dirichlet unit theorem
13. 4. lattices, proof of Minkowski bound
6. 4. Easter
30. 3. ideal norm, Minkowski bound and applications
23. 3. finding prime factorizations, ramification
16. 3. prime decompositions, efg theorem
9. 3. unique factorization into product of ideals (to Sec. 2.6)
2. 3. existence of integral basis (to Sec. 2.2)
23. 2. norm and trace, discriminant (to Prop. 2.5)
16. 2. introduction
23. 3. finding prime factorizations, ramification
16. 3. prime decompositions, efg theorem
9. 3. unique factorization into product of ideals (to Sec. 2.6)
2. 3. existence of integral basis (to Sec. 2.2)
23. 2. norm and trace, discriminant (to Prop. 2.5)
16. 2. introduction
Recommended reading
Main sources
A rough version of my new lecture notes – I suggest you print it maybe only 1-2 weeks ahead of class, probably there’ll be quite a lot of changes.
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84 (Second Edition)
A rough version of my new lecture notes – I suggest you print it maybe only 1-2 weeks ahead of class, probably there’ll be quite a lot of changes.
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84 (Second Edition)
J. Milne, Algebraic Number Theory, http://www.jmilne.org/math/CourseNotes/ant.html
Other sources
Serge Lang, Algebraic Number Theory, GTM 110, 1994.
E.I. Borevič, I.R. Šafarevič: Number Theory, Academic Press 1966.
H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin 1996.
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)
Last year’s course page; this year will be quite similar. Older course pages: 2018/19 (in Czech), 2020/21.
