Topics in Number Theory 2009-2010

Contact information

The exam of 8th June 2010

• A pdf file of the full exam is here.

Rules for homework

• The second and later homework assignments can be done either entirely individually or by working in pairs. In the latter case only one homework is to be handed in, carrying both names. Collaboration except when working in pairs is not allowed.
• You can hand in the homework in the (Tuesday) class. Alternatively, you can either email a copy to Tejaswi at tejaswi@few.vu.nl or (if convenient) put a copy of the homework in the mailbox (of Navilarekallu) in the science building at the VU (Room S-322). If you are submitting by email or in the mailbox, the deadline is 2 p.m. on the due date.
• If you have any questions, email Tejaswi.

The references below are to Gouvêa's book "p-adic Numbers: An Introduction". A small number of typographical errors are listed here.

Date	Material covered	Exercise sheet and homework
9th February	Introduction. Chapter 2 up to Definition 2.3.5 (inclusive).	Exercise sheet 1 (weeks 1-3)
16th February	The remainder of Chapter 2, and Section 3.1.
23rd February	Section 3.2 (summary). Of Section 3.3, essentially 3.3.4, 3.3.11 and 3.3.12.	Assignment 1 (due Wednesday 3rd March) Updated 1/3/2010 to clarify Problem 3 (and to update the number of the room with mailboxes)
2nd March	3.3.6, 3.3.8 and the more sophisticated version of 3.4.1 in Problem 112 (taken from Robert's book A course on p-adic analysis).	Exercise sheet 2 (week 4)
9th March	Stated 3.4.6 and 3.5.2. Discussed Chapter 4 up to 4.3.2 (inclusive). Note: a stronger verion of 4.1.4 was proved. Also, the proofs in this chapter usually work for more general fields k than Qp (e.g., k complete with respect to a non-archimedean absolute value; sometimes we need that k contains Qp and that the absolute value on k restricts to |.|p on Qp).	Exercise sheet 3 (week 5) Assignment 2 (due Tuesday 23rd March)
16th March	4.3.3; 4.4.1, 4.4.3, 4.4.4 and 4.4.5 (in condensed form). 4.5.1, 4.5.3, 4.5.5, 4.5.7, 4.5.8 (in a stronger form), 4.5.9.
23rd March	4.4.6, 4.4.7, 4.4.8, Sections 5.1 and 5.2	Assignment 3 (due Tuesday 6th April) Solutions (added 30/5/2010)
30th March	Most of Section 5.3 up to 5.3.6 (inclusive), but avoiding the use of field automorphisms; of Section 5.4, gave the definitions of e and f, valuation, and uniformizer; stated 5.4.6 (e*f=[E:Qp]) and 5.4.11.	Exercise sheet 4 (week 8)

If you want to read the rest of the book, then the highlights are

• the construction of C_p in Section 5.7, which is both complete (by construction) and algebraically closed (essentially because of Krasner's lemma, 5.7.2), but not locally compact;
• Section 6.4 gives a relation between the sizes of coefficients of polynomials or power series and how many roots there are of a given size, by means of some points and lines in the plane (Newton polygon); the main results here are 6.4.7 (for polynomials) and 6.4.11 (for power series);
• of independent interest are the p-adic Weierstrass preparation theorem, 6.2.6, as well as Krasner's lemma, 5.7.2.