The Hasse–Minkowski Theorem
نویسنده
چکیده
The Hasse-Minkowski Theorem provides a characterization of the rational quadratic forms. What follows is a proof of the Hasse-Minkowski Theorem paraphrased from the book, Number Theory by Z.I. Borevich and I.R. Shafarevich [1]. Throughout this paper, some familiarity with the p-adic numbers and the Hilbert symbol is assumed and some basic facts about quadratic forms are stated without proof. Also, the proof of the Hasse–Minkowski theorem given here uses the Dirichlet theorem on primes in arithmetic progressions. A proof of Dirichlet’s theorem will not be given here (see [1], for a proof of the theorem) due to its length, but the result is stated presently.
منابع مشابه
18.782 Arithmetic Geometry Lecture Note 10
In this lecture we lay the groundwork needed to prove the Hasse-Minkowski theorem for Q, which states that a quadratic form over Q represents 0 if and only if it represents 0 over every completion of Q (as proved by Minkowski). The statement still holds if Q is replaced by any number field (as proved by Hasse), but we will restrict our attention to Q. Unless otherwise indicated, we use p throug...
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