Advanced Algebraic Number Theory NMAL431

Welcome to the Advanced Algebraic Number Theory NMAL431 course webpage for the academic year 2026/27.

About the course

This course is a continuation of Basic Algebraic Number Theory NMAG472 (see Vita Kala’s notes here). The course will broadly follow Jack Thorne’s lecture notes from the Cambridge Part III course in algebraic number theory (found here). The aims of this course are to cover the following:

• Dedekind domains, localisations, and completion at a prime.
• Galois theory of local and global fields.
• An introduction to global class field theory.

When and where

Lectures take place every Tuesday at 12:20 in room K10C, and again at 17:20 in room K7 (90 minutes each). The first lecture will be on 29 September 2026. Every second week, starting 6 October 2026, the 17:20 session will be an exercise class.

There will be no teaching on 17 November 2026.

Exercise sheets

• Sheet 1 (6 October) [questions]

Homework

There are three homework sheets in total. You may hand me your solutions at the beginning of the exercise class on the due date, or place them in my pigeonhole (3rd floor) before the class. Credit for the course (zápočet) will be awarded by receiving a score of at least 50% on each homework set.

• Homework 1 [pdf] (due: 3 November 2026)
• Homework 2 [pdf] (due: 8 December 2026)
• Homework 3 [pdf] (due: 5 January 2027)

Assessment

The assessment for this course is an oral exam, in which you will be asked to prove a theorem from the course. If you would like to take the exam, please email me with a proposed exam date and time. You may do the exam on any day up until 15 September 2027.

References

• J. Thorne, Algebraic Number Theory, lecture notes for Part III, University of Cambridge, 2019 (pdf).
• M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1969. Chapter 9.
• S. Lang, Algebraic Number Theory. Grad. Texts in Math. 110, Springer, 1994.
• J.-P. Serre, Local Fields. Grad. Texts in Math. 67, Springer, 1979.
• H. P. F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory. London Math. Soc. Stud. Texts 50, Cambridge University Press, 2001.

Please feel free to email me with any questions about the course!