2
$\begingroup$

Witt's cancellation theorem states that if $(V_1,q_1)$, $(V_2,q_2)$, $(V_3,q_3)$ are quadratic forms over a field such that $$(V_1,q_1) \perp (V_3,q_3) \cong (V_2,q_2) \perp (V_3,q_3),$$ then $(V_1,q_1) \cong (V_2,q_2)$. The argument when the characteristic of the ground field is not $2$ is easy to find.

It is a widely cited but very rarely proved fact that this theorem also holds over any field of characteristic two. For example, this is claimed in the appendix of Milnor and Husemöller's Symmetric Bilinear Forms, but it is not proved there, nor does it appear to be proved in the references to Bourbaki or Chevalley (as far as I can tell).

I would be grateful for a reference to a proof of this.

  • 0
    Have you checked O'Meara's book? My sample may be biased, but many people I have met say it is comprehensive. Warning: I have not really studied quadratic forms myself. – 2017-01-01
  • 0
    @JyrkiLahtonen O'Meara's book does not discuss characteristic $2$ at all (there is no mention of the Arf invariant in the book, for example) – 2017-01-01
  • 0
    Ok. Sorry. My memory was at fault. I once reviewed a coding theoretical paper using Arf-Kervaire in characteristics four, eight et cetera. I found a not insignifcant part of the result from O'Meara. But I now remember that the argument was 2-adic in nature, and the detail I looked up was then naturally included. It was about sufficient sets of invariants for 2-adic forms. – 2017-01-01
  • 0
    @Jyrki it is in Grove, put answer – 2017-01-01

1 Answers 1

1

Here is a short, recent article that gives examples of failure for Witt cancellation over some rings for which 2 is special in some way. Might take some work to figure out an example over a field of characteristic two... http://users.math.yale.edu/~auel/papers/docs/wittlocal.pdf

For more detail, probably worth writing to Prof. Auel at Yale.

Corollary 12.12 on page 118 of Classical Groups and Geometric Algebra by Larry C. Grove http://bookstore.ams.org/gsm-39

second occurrence at GROVE

enter image description here

  • 0
    Thanks, I am checking it out! There are some additional assumptions in Grove's book (nondefective and ground field is perfect) which is just as well, as I am starting to suspect that the cancellation theorem does not hold as generally as sometimes claimed... For my purposes it should work – 2017-01-01
  • 0
    @user6246 looking in Lam (2004), he recommends Baeza (1978) and Knus(1991), both titles say forms over rings. Probably more difficult than Grove, but might have examples for the necessity of extra assumptions in characteristic two....Grove died in 2006. – 2017-01-01