Syllabus

Math 351, Spring 2007

David S. Dummit

OFFICE: Room 333 Waterman (Dean's Office, Graduate College)
6-5783 (office), 6-2940 (Department office)
e-mail: dummit@math.uvm.edu

OFFICE HOURS: Any time by appointment

TEXT: There is no text for this course, but some suggested references are:

Local Fields, J.W.S. Cassels, London Mathematical Society Student Texts, 3, Cambridge University Press, 1986

Algebraic Number Theory (the "Brighton Conference"), J.W.S. Cassels and A. Frohlich, Academic Press, 1967

p-adic numbers, p-adic analysis and zeta functions, N. Koblitz, Springer, 1984

Local Fields, J.P. Serre, Springer, 1980

OVERVIEW: This course will cover the basic theory of local fields, concentrating on the field of p-adic numbers and their application to problems in group theory and number theory. The course will begin with the proof of a basic theorem (Ostrowski) showing that the real numbers and the p-adic numbers are the only completions of the rational numbers with respect to a metric defined by an absolute value. While the p-adic numbers arise naturally in number theory from considerations modulo p^n for arbitrary n, the fact that the p-adic numbers arise as a completion of the rationals allows analytic tools to enter the picture (where some of your fondest wishes come true: for example infinite series converge if and only if their terms tend to 0). The basic arithmetic and analytic properties of the p-adic numbers will be developed, including the definition of the p-adic integers and the fact that they are a compact, totally disconnected, topological group, with additional topics chosen as interest and enthusiasm dictate (e.g., p-adic properties of elliptic curves, p-adic L-series, Hasse-Minkowski "local-global" theorem for quadratic forms, etc.)

As an advanced graduate course, there will be no exams. Your course grade will be determined on the basis of your solutions to the homework problems; this includes both written solutions and solutions presented in class.

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