Part II Number Fields, Lent 2025

This page contains information for students taking my lectures on Number Fields in Lent 2025. The best way to contact me outside lectures is by email: pv270@dpmms.cam.ac.uk. (I do not monitor Moodle.)

Lecture notes are available here. (Last updated: 20 March 2025.) The notes will be updated in the course of the term. Please check back, and see the changelog below.

Example sheets:

• First example sheet. (4 February 2024.)
• Second example sheet. (19 February 2024.)
• Third example sheet. (3 March 2024.)

Lecture recordings are available via Panopto. However, I expect that most students will find the notes more useful.

Changelogs

Notes

• Version of 19 January 2025: Contains only the first lecture. More to come soon.
• Version of 28 January 2025: Made some minor improvements to the notation in Section 1. Added Sections 2-3 that cover the ring of integers and its additive structure.
• Version of 31 January 2025: Improved the presentation around what is now Lemma 15. Two minor improvements to the presentation in Section 3. (Previous version was correct.)
• Version of 5 February 2025: Added Section 4 (unique factorization of ideals).
• Version of 19 February 2025: Added Sections 5 and 6 (norms of ideals and ideals in field extensions).
• Version of 2 March 2025: Added Sections 7 and first half of 8 (class group and units).
• Version of 10 March 2025: Added Sections 8 and 9 (Dirichlet's unit theorem and Cyclotomic fields with application to FLT).
• Version of 20 March 2025: Some minor improvements of the presentation and some typos corrected. These include: a paragraph in Section 4 about why working with the ring of integers is important, an explanation about extending the defining property of prime ideals to ideals on page 18, the proof of Theorem 35 is now given as Noetherian induction, the discussion in the beginning of Section 6 has been extended to fractional ideals, a brief explanation after Theorem 85 why the two descriptions of roots of unity are the same.