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 prepared lecture notes during the semester. Here’s a draft of the notes (which is not perfect, but I’ve finished the first round of fixes based on the current course).

Office hours

Please email me if you want to discuss anything with me!

Covered material

18. 5. exam date
11. 5. cyclotomic fields and FLT (to end of Ch. 5)
4. 5. cyclotomic fields (most of Sec. 5.1)
27. 4. proof of Dirichlet unit theorem (to end of Ch. 4)
20. 4. proof of Minkowski bound, Dirichlet unit theorem (to L. 4.21)
13. 4. lattices, sum of four squares (Sec. 4.3 and 4.4)
30. 3. ideal norm, Minkowski bound and applications (Sec. 4.1 and 4.2)
23. 3. finding prime factorizations, ramification (Sec. 3.2 and 3.3)
16. 3. prime decompositions, efg theorem (Sec. 2.7 and 3.1)
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

Office hours

Covered material

Recommended reading