From: "Jérôme Dubois" <jdubois@ucfma.univ-bpclermont.fr>
Subject: Re: Quadratic forms
Date: Sun, 18 Feb 2001 17:04:54 +0100
Newsgroups: sci.math
Summary: Structure of Witt Ring W(F) of a field? (classification of quadratic forms)

Azmi Tamid <azmi_tamid@my-deja.com> a écrit dans le message :
mfjt6tpmmmh9@forum.mathforum.com...
> I know that over the reals R two quadratic forms are equivalent iff
> they have the same rank and the same signature.
>
> Is there a similiar theorem saying when two such form are equivalent
> when the field is a finite field ? or maybe other fields.
>
> Thanks,
> Azmi

Hi,
In general and for an arbitrary field this problem is very difficult ; here
some indications :

1) if the field is not of caracteristic 2 and if it is algebracally closed,
then there is just one equivalence class of non degenerated quad forms on it
;

2) if the field is a finite field not of caracteristic 2 k=F_q, then there
is two equivalence classas of non degenerated quad forms and moreover these
classes are caracterised by the discriminant of the form : in one case the
discriminant is a square in F_q^*, and and the other it is not a square ;

3) if the field is perfect and of caracteristic 2 the german mathematician
C. Arf introduce an invariant of quad froms called the Arf invariant and two
non degenerated quad forms are equivalent iff their Arf invariants are the
same. (if you want i can tell you much on the arf invariant just tell me)

regards,
Jerome
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From: dcohen@sophia.inria.fr (David Cohen-Steiner)
Subject: Re: Quadratic forms
Date: 19 Feb 2001 13:28:10 GMT
Newsgroups: sci.math

[quotes of original and followup posts deleted --djr]

A related important conjecture is the so-called Milnor's conjecture.
It has been proved recently by Voevodsky, at least partially.

Given two F-vector spaces V and W endowed with quadratic forms v and w, VxW is
naturally endowed with the following quadratic form :
((x1,y1),(x2,y2)) --> v(x1,y1) + w(x2,y2)

Calling it the sum of v and w, one gets a monoid structure for the set
of quadratic forms.

Now add formal inverses (just like for building Z from N) : you get a group.

To be precise, one usually considers the quotient of this group by
totally isotropic forms, and one only considers equivalence classes of
q.f.(the sum does not depend on the chosen member).

The tensor product of vector spaces gives rise to a product on the
q.f. The q.f. thus now form an algebra. The Milnor conjecture states
that this algebra is isomorphic to two others :

- The Milnor algebra, which is additively isomorphic to (F^*,x) and whose 
  multiplication is tensor product modulo a simple relation (a tens (1-a) = 0)

- The cohomology algebra of the absolute Galois group of F with coefficients 
  in Z/2.

A search in arxiv should give you more details ("Voevodsky" for example)

Regards,

David
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[Both messages reformatted --djr