Quadratic forms over dyadic valued fields. II, relative rigidity and Galois cohomology

Quadratic Forms over Arbitrary Fields

Introduction. Witt [5] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only...

Quadratic Forms over Global Fields

1. The Hasse Principle(s) For Quadratic Forms Over Global Fields 1 1.1. Reminders on global fields 1 1.2. Statement of the Hasse Principles 2 2. The Hasse Principle Over Q 3 2.1. Preliminary Results: Reciprocity and Approximation 3 2.2. n ≤ 1 6 2.3. n = 2 6 2.4. n = 3 6 2.5. n = 4 8 2.6. n ≥ 5 9 3. The Hasse Principle Over a Global Field 9 3.1. n = 2 10 3.2. n = 3 10 3.3. n = 4 11 3.4. n ≥ 5 12...

Quadratic Forms over Fields Ii: Structure of the Witt Ring

Pairs of Quadratic Forms over Finite Fields

Let Fq be a finite field with q elements and let X be a set of matrices over Fq. The main results of this paper are explicit expressions for the number of pairs (A,B) of matrices in X such that A has rank r, B has rank s, and A + B has rank k in the cases that (i) X is the set of alternating matrices over Fq and (ii) X is the set of symmetric matrices over Fq for odd q. Our motivation to study ...

Perfect unary forms over real quadratic fields

Let F = Q( √ d) be a real quadratic field with ring of integers O. In this paper we analyze the number hd of GL1(O)orbits of homothety classes of perfect unary forms over F as a function of d. We compute hd exactly for square-free d ≤ 200000. By relating perfect forms to continued fractions, we give bounds on hd and address some questions raised by Watanabe, Yano, and Hayashi.