Fall 2002–2003Ma 160a - Number Theory |
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coursework page. Course Textbook: A Course in Arithmetic, J.P. Serre, Springer (GTM 7). Course Description: This is the first term of a three-term sequence on number theory. This term's course will focus on the theory of quadratic forms, following the treatment given in the "Algebraic Methods" section of Serre's book. We will define the p-adic numbers, and then study quadratic forms over the p-adic fields. We will then prove a "local-global principle," the Hasse-Minkowski Theorem, which will allow us to use our knowledge of quadratic forms over the "local" fields of the p-adic numbers to classify quadratic forms over the "global" field of the rational numbers. We will finally consider the integral quadratic forms — those defined over the rational integers. As a brief digression, we will (time allowing) consider quadratic forms and their use in the theory of classical modular forms. Prerequisite: The only prerequisite for this course is Ma 5. We will cover any additional background topics that arise. Grading: There will be weekly homework worth 50% of the grade and a final exam worth 50%. Collaboration on the homework is allowed, but you must write up your solutions on your own and in your own words. No collaboration will be allowed on the final exam. |