# quadratic forms over fields of characteristic 2

I was wondering if anyone knows any good sources for the theory of quadratic forms over fields of characteristic 2 which are written in English?

• What have you done so far? Typing the title into google gives a lot of results. So what? Mar 20 '12 at 13:02
• Indeed. For instance, you could look at this: jstor.org/stable/2372942 Mar 20 '12 at 13:04
• While indeed you could have put more work into this question, I feel obligated to mention that maybe the biggest tool in the study of quadratic forms over a field of char $\ne 2$ is the bijection with certain bilinear forms. Over a field of char $=2$ there can be many quadratic forms to a single bilinear form. Therefore, one surprising place to start if you know some alg. geom is the theory of theta characteristics, where you fix the Weil pairing on an abelian variety as your bilinear form. Mar 20 '12 at 13:53
• I added the obvious tag "quadratic-forms". Mar 21 '12 at 9:45

This book by Manfred Knebusch starts with the limerick

$$\begin{array}{l}\text{A Mathematician Said Who}\cr\text{Can Quote Me a Theorem that’s True?}\cr\text{For the ones that I Know}\cr\text{Are Simply not So,}\cr\text{When the Characteristic is Two!}\end{array}$$

It gives a uniform treatment of quadratic forms in all characteristics including two.

• Sorry for the strange formatting, I don't know how to get around that... Mar 21 '12 at 0:12
• Like this. :) Mar 21 '12 at 0:48
• (two asterisks on each side of the word) Mar 21 '12 at 0:49
• Irving Kaplansky liked to quote Marshall Hall: "The trouble with two is not that it's so small. It's that it's so even." Mar 21 '12 at 0:52
• The link is now maths.ucd.ie/~tpunger/papers/book.pdf Feb 10 '18 at 19:05

The Algebraic and Geometric Theory of Quadratic Forms by Elman, Karpenko and Merkurjev is a standard recent reference for the theory of quadratic forms, paying special attention to the differences between the theory of bilinear forms and the theory of quadratic forms in characteristic 2.

http://en.wikipedia.org/wiki/Arf_invariant would be a start.