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I'd like to know why all quadratic forms satisfy the following:

1) If $F$ is an algebraically closed field, for example, the field of complex numbers, and $(V, q)$ is a quadratic space of dimension at least two, then it is isotropic.

And:

2) If $F$ is the field $Q_p$ of p-adic numbers and $(V, q)$ is a quadratic space of dimension at least five, then it is isotropic.

Any insight on these two is greatly appreciated.

Adar Gutman
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  • recommend Cassels, inexpensive http://store.doverpublications.com/0486466701.html – Will Jagy Sep 04 '17 at 20:43
  • You need to state clearly what is $(V,q)$ and what means isotropic. Possibly it reduces to every positive definite matrix is diagonalizable over the algebraic closure. – reuns Sep 04 '17 at 21:14
  • Isotropic means nontrivial kernel and $(V, q)$ is simply the quadratic form $q:V \rightarrow F$ – Adar Gutman Sep 04 '17 at 21:41

1 Answers1

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On the first one, from Cassels. He has characteristic not two. He does not say that all forms can be diagonalised over a field, he says that there is a normal basis. We use the polarization identity to get an inner product $\phi,$ a normal basis has $\phi(u_i, u_j) = 0$ when $i \neq j.$ Therefore the form applied to a vector becomes $\sum a_i x_i^2.$ Over an algebraically closed field, we can take all but the first two $x_i$ zero, then take $x_2 = 1,$ and solve for $x_1$ in $a_1 x_1^2 + a_2 = 0.$

The dimension 5 bit is a longer story. I recommend Cassels because he is focused on the rational numbers and the rational integers, he does isotropy for the $p$-adics in great and understandable detail. This includes tables for the Hilbert norm residue symbol, pages 43 and 44, with which the isotropy results are proved, pages 58-60 primarily. Your question is Lemma 2.7, page 60. The proof is pigeonhole on the $p$-adic squareclasses

Will Jagy
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  • @reuns yes, my $x_i = \langle x,u_i\rangle.$ Cassels does use the word diagonal later in the book. – Will Jagy Sep 04 '17 at 21:22
  • So $\sum_i a_i \langle x,u_i\rangle^2$ the diagonalized form. And what is special over $\mathbf{Q}_p$ (and $\dim(V) \ge 5$) ? – reuns Sep 04 '17 at 21:24
  • @reuns, he shows that any ternary represents all but one p-adic squareclass. The remaining binary part (out of dimension 5), rather $-1$ times the binary section, represents half the squareclasses. Thus there is a nonzero element represented by both pieces. Hmmm, then he says he has already proved it for prime $p \neq 2$ in Lemma 1.7, Corollary, of Chapter 3, page 41. Good to know. – Will Jagy Sep 04 '17 at 21:29
  • Tks, your answer and this gives some clues – reuns Sep 04 '17 at 21:40