### NUMBER THEORY III/IV,   Michaelmas Term 2011

Lectures:

• Monday 9 in CM107
• Tuesday 9 in CM107  no lecture on Tue 22.02.
• Problems class: Wednesday, 16.02. at 11am in E102.
• Second problems class: Wednesday, 09.03. at 11am in CG218
• Extra lecture: Wednesday, 16.03. at 11am in CG218

Notes: Here are improved notes for the course, covering the Michaelmas term and most of the Epiphany term.

Moreover, in case you missed a lecture, there are scans from the lectures of Michaelmas term and of Part 1 and Part 2 of Epiphany term, courtesy of Natasha Morrison.

Problems:

 (from Mmas) some solutions to Problem Sheet 2 [pdf] (from Mmas) some solutions to Problem Sheet 3 [pdf] (from Mmas) some solutions to Problem Sheet 4 [pdf] (from Mmas) Problem Sheet 5 [pdf] Solutions (from Mmas) Problem Sheet 6 [pdf] Solutions Note: The discriminant in Sheet 10, Q4, is 404, rather than 2^8*3^2 (spotted by Natasha).

Challenges:

First challenge (solved by Natasha Morrison, Steven Charlton and Matthew Palmer); here is Matt's very nice solution.

Second challenge (results from Robert Botcherby, Alex Kent, Caspar de Haes, Nick Byrne and Steven Charlton); here is a link to Steven's outstanding families of solutions!

There is actually a rather recent paper dealing with the background (and solution) of the problem (link provided by Natasha).

Reading material for year IV students:

Appendices A and B of Stewart-Tall Algebraic number theory and Fermat's last theorem .

Here is now a library copy of Appendix A for module MATH4211 Number Theory IV (DUO password needed), and here is a similar link for interested 3rd year students, i.e. module MATH3031 Number Theory III.

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.

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 summaries/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 last year's course ANT II.

Assignments:  Set every fortnight (roughly); collected and marked only twice per term (presumably in week 4 and 8). Will be handed out successively and will then also be available here.

To hand in by Feb 08: Sheet 7, Q1), Q3), Q6)(i), (ii).

To hand in by Mar 08: Sheet 8, Q2 i), Q5 a), c), Q7); and Sheet 9, Q2) for d=-21.