NUMBER THEORY III/IV,   Michaelmas Term 2012


The 'Performance':

Wrapping up the course: Lyrics of the short performance during the last lecture.

Here is a short music clip indicating how it should have sounded.


NEW: Lecture notes, with some further background material, are given here (latest version, 46 pages, updated Dec 28). Thanks for corrections/comments to Paul Shearer, Aaron Paxton, James Mankelow.

Here are notes for a similar previous course, then covering the Michaelmas term and most of the Epiphany term.

Moreover, there are scans from the lectures of Michaelmas term (given by a different lecturer) and of Part 1 and Part 2 of Epiphany term, courtesy of Natasha Morrison.

As indicated in the lectures, here is a one-sentence proof that each prime congruent to 1 modulo 4 is a sum of two squares.


A new sheet given every fortnight (roughly); collected and marked only twice per term (presumably in week 4 and 8). Will be handed out successively and will thereafter (or possibly even before) also be available here.

Problem Sheet 1 [pdf] A few hints [pdf] Solutions [pdf]

Problem Sheet 2 [pdf] Solutions [pdf]

Problem Sheet 3 [pdf] Solutions [pdf]

Problem Sheet 4 [pdf] (most) Solutions [pdf]

Problem Sheet 5 [pdf] (most) Solutions [pdf]

Problem Sheet 6 [pdf] (most) Solutions [pdf]

Correction for Q.2(b) on Prob Sheet 3: in eq.(1) need the modulus: |phi(beta*gamma)| > |phi(gamma)| [spotted by Aaron Paxton].

Correction for Q.4(ii) on Prob Sheet 4: the ideal on the RHS should be (1+theta) rather than (1-theta) [spotted by Serena Liu and by Zhe Chen].

To hand in by Tue 6.11. in the lecture: Sheet 2, Q1), Q10), Q12), Q5(ii).

To hand in by Thu 6.12. in the Maths Office (separate folders for NT III and NT IV): from Sheet 4, Q2), Q6(iii), Q7) and from Sheet 5, Q2(i),Q2(vii).

Extra reading material (concerns year 4 students mainly):

Extra reading for fourth year and MSc students is Chapter 9 of Ireland and Rosen, on cubic and biquadratic reciprocity, the link has been emailed to the class on 13 Dec. There will be one mandatory exam question concerning that material, worth 20 marks; this question will constitute Section C of their exam. (Year 3 students are invited to read that chapter, too, but it will not be examinable for them.) Facts used in this chapter referring to previous ones can be taken as true (i.e. no proofs thereof will be asked in the exam).

Click here for an account of the statements that are being referred to.

Reading suggestions:

Each of the following references covers most of the material in the lectures (in rather different form). Examples and motivation are taken from various sources. A good compilation of most of the material of the course (and far beyond) can be found in

  • Ian Stewart, David Tall: Algebraic number theory and Fermat's last theorem (A K Peters)
  • Daniel A. Marcus: Number theory (Springer Universitext)
  • both of which furthermore providing lots of good exercises.

  • Ian Stewart: Galois theory Chapman and Hall
  • has been recommended by a student in class as it covers many of the topics from the lectures.

    A famous classic is

  • Senon Borewicz and Igor Safarevic: Number theory (Academic Press)
  • Here is a more recent standard reference, from a somewhat more advanced point of view

  • Jürgen Neukirch: Algebraic number theory (Springer)
  • The following online notes of a course given by Milne contain a wealth of material.

  • James Milne: Algebraic Number Theory (Course notes available online)
  • Lots of details are explained in

  • Paulo Ribenboim: Algebraic numbers (Wiley-Interscience)
  • while a very concise account of the theory can be found in

  • Sir Peter Swinnerton-Dyer: A Brief Guide to Algebraic Number Theory (LMS Student Texts)
  • For exciting piece-wise (elementary) reading, with mostly independent sections in historical order, check out

  • Winfried Scharlau, Hans Opolka: From Fermat to Minkowski (Springer Undergraduate Texts)

  • Furthermore, here you can find a large library of links (some are already outdated, alas) to online notes (some of them in French, German, Dutch or even Greek!).

    Memory "refreshments":

    There are scans of notes taken by Steven Charlton for the first term (summary and scans (part 1), scans (part 2)) and for the second term (summary and scans (part 3), scans (part 4)) of a course Algebra and Number Theory II which covers much of last year's Algebra material (and a bit beyond), and is a good summary of background material.

    For material on divisibility and congruences, you can refer to notes from last year's course ENTC2.