# Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont.

First we asumme $char(k) \neq 2$ for a non specific fiel $k$. Any quadratic form over $k$ can be diagonalized,meaning we can write it as $q = <a_1,..,a_n> =a_1X_1^2 +...+a_nX_n^2$. We denote by $W(k)$ the Witt-Ring of isometric classes of forms over $k$.

There are three "classic invariants",as Arason calls them in his dissertation $Cohomologische Invarianten Quadratischer Formen$.

1. $dim: W(k) -> \mathbb{Z}/2\mathbb{Z}$ dimensionindex
2. $disc: W(k) -> k^*/k^{*2}$ the discriminant.
3. $\epsilon: W(k) -> Br_2(k)$ the Hasse-Invariant/Clifford Invariant.

In our case $disc(q)$ is just squareclass of the determinant of the diagonalmatrix $A_q$ of $q$.

Let $I=ker(dim)$ be the fundamental ideal of $W(k)$.

Arason proves: $dim,disc,\epsilon$ allow a complete a complete classification of the elements of $W(k)$ iff $I^3=0$.

A problem is,that disc and $\epsilon$ are not homomorphisms. So we need to restrict them. To see what would be useful to restrict the maps to some basic calculations suffice. So $disc$ is restricted to $I=ker(dim)$ and $\epsilon$ to $I^2=ker(disc)$. (the kernels are harder to calculate!).

The maps we now obtain are not $disc$ and $\epsilon$ anymore but they now are called $e_1$ and $e_2$, while $e_0:= dim$.

The Arason-Pfister Hauptsatz states that every potence $I^n$ is additively generated by $n-fold$ Pfisterforms and every anisotropic element in $I^n$ hast at least dimension $2^n$.

Last fact: We have a filtration $W(k) \supset I \supset I^2..$

Here are the questions.

1.Since there are only elements of even dimension in every $I^n$, how can i calculate $e_1,e_2$ of odd dimensional ones?

2.This question might be analog to: Do $e_0,e_1,e_2$ also discribe the whole Wittring if $I^3$ vanishes?

Let $p \neq 2$.

3 The discriminant of certain forms might be negative. Also when restricted to even dimensional forms. Take X^2-Y^2 for example. But in case of $k = \mathbb{Q}_p$ the squareclasses are represented by ${1,u,p,up}$ with $u$ beeing a non square mod $p$ and $-1$ is only a square in $\mathbb{Q}_p$ iff $p=4k+1$. I dont see why the discriminant is well defined in these cases.

4 Let $u,p,up$ be as in 3. They induce Pfisterforms $<<u>>$ etc. Is the tensorproduct -square of these equal to zero i.e does $<<u>> \otimes <<u>>= 0$ hold? Can one tell without knowing $p$ ?