%%% X-UIDL: 17ae4163db21b9be0e59cf6992d32676
%%% Date: Tue, 3 Oct 2000 16:51:26 +0200 (CEST)
%%% From: Ahmed LAGHRIBI <laghribi@euler.univ-artois.fr>
%%% To: rehmann@mathematik.Uni-Bielefeld.DE
%%% Subject: Re: A note on the generic splitting
%%% In-Reply-To: <200009281700.TAA05068@ada.mathematik.uni-bielefeld.de>
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%%% 
%%% 
%%% 
%%% Dear Professor Rehmann,
%%% 
%%% Here is the latex file of my preprint "On the generic splitting of
%%% quadratic forms in charactersitic 2" that I would like to publish on the
%%% LAG sever. Because this preprint is already submitted to a Journal I don't
%%% known if its publication on the LAG sever is accepted by this journal or
%%% no. Perhaps I must ask the editor of this journal?
%%% 
%%% Best regards,
%%% 
%%% Ahmed 
%%% 
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\begin{document}

\title[On the generic splitting of quadratic forms $\cdots$]{On the
generic
splitting of quadratic forms in characteristic 2}
\author[A. Laghribi]{Ahmed Laghribi}
\address{Facult\'e Jean Perrin \\ Rue Jean Souvraz - SP 18\\ 62307
Lens\\France}
\email{laghribi@@euler.univ-artois.fr}
\thanks{1991 Mathematics Subject Classification: 11E04, 11E81\\
Keywords and Phrases: Quadratic Form, Function Field of a Quadric,
Generic Splitting of a Quadratic Form, Pfister Form, Pfister
Neighbor.}
%\date{\today}
\begin{abstract}
{In \cite{knebusch1} and \cite{knebusch2} Knebusch established the
basic facts of generic splitting theory of quadratic forms over a
field of characteristic not $2$. In \cite{knebusch3}, he
generalized this theory to a field of characteristic $2$. This
note is related to \cite{knebusch3}. More precisely, we begin with
a complete characterization of quadratic forms of height $1$ (in
this note we don't exclude anisotropic quadratic forms with
radical of dimension at least $1$). This allows us to extend the
notion of degree in characteristic $2$. We prove some results on 
excellent forms and generic splitting tower of a quadratic form. Some 
results on quadratic forms of height $2$ and degree $1$ or $2$ are given.}
\end{abstract}
\maketitle
\small
\begin{sloppypar}
Let $F$ be a field of characteristic not $2$. We associate to an
anisotropic quadratic form $\phi$ of dimension $\geq 3$ over $F$,
the function field $F(\phi)$ of the projective quadric defined by
the equation $\phi=0$. When $\phi$ is an anisotropic quadratic
form of dimension $2$ (resp. of dimension $1$ or isotropic of
dimension $2$) we set $F(\phi)=F(\sqrt {-\det \phi})$ where $\det
\phi$ is the discriminant of $\phi$ (resp. $F(\phi)=F$). An
anisotropic quadratic form of dimension at least $2$ becomes
clearly isotropic over its function field. In \cite{knebusch1}
Knebusch associated to a non-split quadratic form $\phi$ a
sequence of extensions and quadratic forms, {\it the so-called
generic splitting tower of $\phi$}, as follows: $\phi_0=\phi_{an}$
(the anisotropic part of $\phi$), $F_0=F$ and inductively for
$n\geq 1$, $F_n=F_{n-1}(\phi_{n-1})$ and
$\phi_n=((\phi_{n-1})_{F_{n}})_{an}$. The height of $\phi$ is the
smallest integer $h=\h (\phi)$ such that $\dim \phi_h \leq 1$.
Clearly, for an anisotropic quadratic form $\phi$ of dimension at
least $2$, we have $\h (\phi)=\h (\phi_{F(\phi)})+1$. In
particular, an anisotropic quadratic form $\phi$ of dimension
$\geq 2$ is of height $1$ if and only if $\dim
(\phi_{F(\phi)})_{an} \leq 1$. For any details on generic
splitting theory of quadratic forms in characteristic not $2$, we
refer to Knebusch's papers \cite{knebusch1}, \cite{knebusch2}.\\
\end{sloppypar}
\begin{sloppypar}
A quadratic form $\phi$ is called a Pfister neighbor if there
exists an $n$-fold Pfister form $\pi$ and a scalar $a\in F^*$ such
that $\dim\phi >2^{n-1}$ and $a\pi \cong \phi \perp \psi$ for some
quadratic form $\psi$, where $\cong$ and $\perp$ denote
respectively the isometry and orthogonal sum of quadratic forms.
The forms $\pi$ and $\psi$ are unique up to isometry. We call
$\psi$ the complementary form of $\phi$ and $\dim \pi -\dim \phi$
the codimension of $\phi$.\\

In general, for a given integer $h$ it is very difficult to
describe quadratic forms of height $h$. In characteristic not $2$,
Knebusch \cite{knebusch1} and Wadsworth \cite{wadsworth} have
given independently a complete characterization of anisotropic
quadratic forms of height $1$. Such a quadratic form is a Pfister
neighbor of codimension $0$ or $1$.\\

In \cite{knebusch3} Knebusch generalized the generic splitting
theory of quadratic forms to a field of characteristic $2$. The
height and the generic splitting are defined in the same manner as
in characteristic not $2$.\\
\end{sloppypar}
\begin{sloppypar}
%%% FROM PROBLEM:
%>From now on, we assume that $F$ is a field of characteristic $2$.
From now on, we assume that $F$ is a field of characteristic $2$.
We will investigate the generic splitting of quadratic forms over
$F$. Along this note, we don't exclude quadratic forms with
radical of dimension at least $1$. We begin with a complete
characterization of anisotropic quadratic forms of height $1$. We
use that characterization to extend the notion of degree in
charactersitic $2$, and we prove some results on anisotropic
quadratic forms of height $2$ and degree $1$ or $2$ like those
obtained in characteristic not $2$ \cite{fitzgerald},
\cite{knebusch2}, \cite{kahn}. We extend the notion of excellent
form and we give some results related to those forms. Some general
results on the generic splitting tower of a quadratic form are
given.\\
\end{sloppypar}
\begin{sloppypar}
All basic facts and details on quadratic forms in characteristic
$2$ can be found in Baeza's book \cite{b1}. Let $\left[ a,
b\right]$ (resp. $\left[ a\right]$) denote the quadratic form
$aX^2+ XY +bY^2$ (resp. the quadratic form $aX^2$).\\

Every quadratic form $\phi$ of dimension at least $1$ can be
written up to isometry:
\begin{equation}
\phi = [a_1, b_1] \perp \cdots \perp [a_r, b_r]
\perp [c_1] \perp
\cdots \perp [c_s] \perp [0]\perp \cdots
\perp [0]
\label{eme}
\end{equation}
with $[c_1] \perp \cdots \perp [c_s]$ anisotropic.
\begin{definition} Let $\phi$ be as in equation (\ref{eme}).\\
(1) If $\dim \phi=2r+s$, then $\phi$ is called a regular form or a
form of type $(r, s)$.\\ (2) A quadratic form of type $(r, 0)$ is
called a nonsingular quadratic form.\\ (3) A quadratic form of
type $(r, s)$ with $s\geq 1$ is called a singular quadratic
form.\\ (4) A quadratic form of type $(0, s)$ is called a totally
singular quadratic form.\\ (5) The quadratic form $[c_1] \perp
\cdots \perp [c_s] \perp [0]\perp \cdots \perp [0]$ is called the
quasilinear part of $\phi$.
\end{definition}
Let $W_q(F)$ denote the Witt group of nonsingular quadratic forms,
and let $W(F)$ denote the Witt ring of nonsingular bilinear
symmetric forms. It is well known that $W_q(F)$ is a
$W(F)$-module. \vskip1mm We denote by $\left< a_1, \cdots,
a_n\right>$ the bilinear form $\sum\limits_{i=1}^n a_iX_iY_i$. For
an integer $r$ and a quadratic form $\phi$, we denote by $r\times
\phi$ the quadratic form ${\underbrace{ \phi \perp \cdots \perp
\phi}_{r \tim }}$. A quadratic form $\phi$ is called split if
$\phi\cong r\times \H \perp s\times \left[ 0\right]$ where
$\H=\left[ 0, 0\right]$ is the hyperbolic plane. If $\phi$ and
$\psi$ are quadratic forms, then $\phi \sim \psi$ means that $\phi
\perp r\times \H \cong \psi \perp s\times \H$. We say that $\phi$
and $\psi$ are similar if $\phi \cong a\psi$ for some $a\in F^*$.
A nonsingular quadratic form $\phi$ is hyperbolic if $\phi \cong
r\times \H$. For a field extension $K/F$ and a quadratic form
$\phi$ over $F$, the quadratic form $\phi\otimes K$ is denoted by
$\phi_K$.\\

An $n$-fold Pfister form is a quadratic form $\left<1,
a_1\right>\otimes \cdots \otimes\left<1, a_n\right>\otimes [1,
a_{n+1}]$ that we denote by $\left< \left< a_1 , \cdots,
a_{n+1}\right] \right]$. Let $P_nF$ (resp. $GP_nF$) denote the set
of $n$-fold Pfister forms (resp. the set $\{ \alpha \pi \mid
\alpha \in F^*, \pi \in P_nF\}$). An $n$-fold Pfister form $\pi$
is either anisotropic or hyperbolic \cite[Chapter 4, Corollary
3.2]{b1}. The $W(F)$-submodule of $W_q(F)$ generated by $n$-fold
Pfister forms is denoted by $I^nW_q(F)$. Let $\br (F)$ denote the
Brauer group of $F$.\\

Set $\pw(x)=x^2+x$ for $x\in F$ and $\pw(F)=\{ \pw(x) \mid x\in F
\}$. If $\phi$ is nonsingular, then the Clifford algebra $C(\phi)$
is a central simple algebra over $F$, and the center $Z(\phi)$ of
the even Clifford algebra $C_0(\phi)$ is a separable quadratic
algebra over $F$. In this case, $Z(\phi)=F(\pw^{-1}(\delta))$ for
some $\delta \in F$ and the Arf invariant $\tr (\phi)$ of $\phi$
is defined as the class of $\delta$ in $F/\pw (F)$. More
precisely, if $\phi \cong a_1[1, b_1] \perp \cdots \perp a_r[1,
b_r]$ then $\tr (\phi)=b_1 +\cdots +b_r \in F/\pw(F)$.\\

Let $\phi$ be a quadratic form of dimension $n \geq 1$ which is
not isometric to $s\times \left[ 0\right]$, and let $P_{\phi}$ be
the homogenous polynomial given by $\phi$. In \cite{ta} it is
shown that $P_{\phi}$ is reducible if and only if $\phi$ is either
isometric to $\H \perp \left[ 0\right]\perp \cdots \perp \left[
0\right]$ or isometric to $\left[ a\right] \perp \left[
0\right]\perp \cdots \perp \left[ 0\right]$ for some $a\in F^*$.
If $P_{\phi}$ is irreducible, we define the function field
$F(\phi)$ of $\phi$ as the field of fractions of $${{F\left[ X_1,
\cdots, X_n\right] }\over {(P_{\phi})}}$$ where $(P_{\phi})$ is
the ideal of $F\left[ X_1, \cdots, X_n\right]$ generated by
$P_{\phi}$. In particular, $F(\phi)$ is well defined for an
anisotropic quadratic form $\phi$. \vskip2mm Quadratic forms of
height $1$ are as follows.
\end{sloppypar}

\begin{theorem}
Let $F$ be a field of characteristic $2$, and let $\phi$ be an
anisotropic quadratic form, possibly singular, of dimension $\geq
1$. Then, $\phi$ is of height $1$ if and only if $\phi$ is one of
the following types:\\ (1) $\dim \phi = 2$,\\ (2) $\phi \in GP_nF$
for some integer $n\geq 1$,\\ (3) $\phi \cong \psi \perp
\left[c\right]$ for some anisotropic $\psi \in W_q(F)$ and $c\in
F^*$ such that $\psi \perp c\left[1, \tr (\psi)\right]\in GP_mF$
for some integer $m \geq 1$.\cn \label{tp}
\end{theorem}
We will need frequently a generalization of the subform notion.
\begin{definition}(\cite{lagh1})
Let $\psi=[a_1, b_1] \perp \cdots \perp [a_r, b_r] \perp [c_1]
\perp \cdots \perp [c_s]$ be a quadratic form.\\ (1) We say that
$\psi$ is dominated by $\phi$ and denote $\psi\prec \phi$ if there
exists a form $\delta$ such that $$\phi \cong [a_1, b_1] \perp
\cdots \perp [a_r, b_r] \perp \xi_1 \perp \cdots \perp \xi_s \perp
\delta$$and for all $i\in \{ 1, \cdots, s\}$, we have
$\xi_i=\left[ c_i\right]$ or $\xi_i=\left[ c_i, d_i\right]$ for
some $d_i \in F$.\\ (2) We say that $\psi$ is weakly dominated by
$\phi$ if $a\psi \prec \phi$ for some scalar $a\in F^*$.\\ (3) We
say that $\psi$ is a subform of $\phi$ and denote $\psi <\phi$ if
$\phi \cong \psi \perp \mu$ for some quadratic form $\mu$.\cn
\label{dd1}
\end{definition}
%%% FROM PROBLEM:
%>From Definition \ref{dd1} we deduce the following remarks.
From Definition \ref{dd1} we deduce the following remarks.

\begin{remar}\
(1) If $\psi$ is weakly dominated by $\phi$ and if $F(\psi)$ is
well defined, then $\phi_{F(\psi)}$ is isotropic.\\ (2) If $\phi$
and $\psi$ are nonsingular and $\psi$ is dominated by $\phi$, then
$\psi$ is a subform of $\phi$.\\ (3) With the same notations and
hypothesis as in Definition \ref{dd1} (1) and if $\phi$ is
nonsingular, then $\dim \xi_i=2$ for all $i\in \{ 1, \cdots, s\}$.
\end{remar}

In \cite{lagh1}, an analogue of the Cassels-Pfister subform
theorem was proved.
\begin{prop}(\cite[Proposition 3.4]{lagh1})
Let $F$ be a field of characteristic $2$, $\phi \in W_q(F)$
anisotropic and $\psi$ be an anisotropic quadratic form, possibly
singular, such that $\phi_{F(\psi)}$ is hyperbolic. Then, $\psi$
is weakly dominated by $\phi$. In particular, $\dim \phi \geq \dim
\psi$. \label{lpp}
\end{prop}
Proposition \ref{lpp} was obtained by using a result of Baeza
concerning the norm theorem for nonsingular quadratic forms
\cite{b2} and some results of \cite{b3}.\\

The following Corollary is an immediate consequence of Proposition
\ref{lpp}.
\begin{cor}
Let $F$ be a field of characteristic $2$, $\phi \in W_q(F)$
anisotropic and $\pi$ be an anisotropic Pfister form. If
$\phi_{F(\pi)}$ is hyperbolic, then there exists a bilinear form
$\rho$ such that $\phi \cong \rho\otimes\pi$. \label{caa}
\end{cor}
The following Lemma is well known, we recall it without proof.
\begin{sloppypar}
\begin{lemma}
Let $F$ be a field of characteristic $2$, $\phi, \psi \in W_q(F)$
and $u_1, \cdots , u_n, v_1, \cdots , v_n \in F$ such that $\phi
\perp \left[ u_1\right] \perp \cdots \perp \left[u_n\right] \cong
\psi \perp \left[ v_1\right] \perp \cdots \perp \left[v_n\right]$.
Then the sets $\{ u_1, \cdots ,u_n \}$, $\{ v_1, \cdots ,v_n \}$
engender the same vector space over $F^2$.\label{luf}
\end{lemma}
\end{sloppypar}
We will need a generalization of the Witt cancellation.
\begin{prop}(\cite[Proposition 1.2]{kn1})
Let $F$ be a field of characteristic $2$, $\eta \in W_q(F)$ and
$\phi,$ $\psi$ be quadratic forms, possibly singular, such that
$\phi \perp \eta \cong \psi \perp \eta$. Then, $\phi \cong \psi$.
\label{gwc}
\end{prop}
By using Proposition \ref{gwc} it is clear that a nonsingular
quadratic form $\phi$ is hyperbolic if and only if $\phi \sim 0$.
\vskip2mm \noindent{{\it Proof of Theorem \ref{tp}.}} To prove
this theorem in the nonsingular case, we use the same idea as in
characteristic not $2$ but we proceed differently by using a
function field argument. The singular case needs some 
observations, the point is that we can't follow directly the proof
given in characteristic not $2$ because Knebusch used a Witt
cancellation argument which is not allowed in characteristic
$2$.\\

We recall a lemma.

\begin{lemma}(\cite[Lemma 3.1]{lagh1})
Let $\phi=\left[ a, b\right] \perp \left[ y\right]$ be a quadratic
form. We have:\\ (1) If $z\in F$ is represented by $\phi$, then
either $\left[ y\right] \cong \left[ z\right]$ or there exists $r
\in  F$ such that $\phi\cong \left[ z, r\right] \perp \left[
y\right]$.\\ (2)  If $\phi$ is isotropic of type $(1, 1)$, then
$\phi\cong \H \perp \left[ y\right]$.\\ (3) If $a=y$, then $\phi
\sim \left[ y\right]$.\label{l1}
\end{lemma}

Let $\phi$ be an anisotropic quadratic form of type (1), (2) or
(3) as in Theorem \ref{tp}. It is clear that $\h (\phi) =1$ when
$\dim \phi =2$ or $\phi \in GP_nF$ for some integer $n\geq 1$.
Let's assume now that there exists a nonsingular quadratic form
$\xi$ and a scalar $c\in F^*$ such that $\phi \cong \xi \perp
\left[ c\right]$ and $\pi:=\xi \perp c\left[ 1, \tr (\xi)\right]
\in GP_mF$ for some integer $m\geq 1$. Since $\phi$ is weakly
dominated by $\pi$, it follows that $\pi_{F(\phi)} \sim 0$ and
$\xi_{F(\phi)} \sim (c\left[ 1, \tr (\xi)\right])_{F(\phi)}$. In
particular, $\phi_{F(\phi)} \sim (c\left[ 1, \tr (\xi)\right]
\perp \left[ c\right])_{F(\phi)}$. By Lemma \ref{l1} and
Proposition \ref{gwc}, we deduce that $(\phi_{F(\phi)})_{an} \cong
(\left[ c\right])_{F(\phi)}$. \vskip2mm For the converse, we may
assume that $\dim \phi \geq 3$ and $\h (\phi) =1$. Set
$\phi=\left[ a_1, b_1\right] \perp \cdots \perp \left[a_r,
b_r\right] \perp \left[ c_1\right] \perp \cdots \perp \left[
c_s\right]$.\vskip2mm \noindent{(1) If $\phi$ is nonsingular: Let
$k$ be the maximal integer such that there exists a $k$-fold
Pfister form $\psi$ as a subform of $\phi$. We have to prove 
$\dim \phi=2^{k+1}$. Assume $\dim \psi <\dim \phi$, and let
$\psi'$ be a quadratic form such that $\phi \cong \psi \perp
\psi'$. Since $\psi$ is anisotropic and $\phi$ is nonsingular of
dimension $\geq 4$, it follows that $F(\psi)(\phi)$ is well
defined. However, $\phi_{F(\psi)(\phi)} \sim \psi'_{F(\psi)(\phi)}
\sim 0$ and $\dim \psi'<\dim \phi$. Proposition \ref{lpp} implies
that $\psi'_{F(\psi)} \sim 0$, and thus $\psi$ is weakly dominated
by $\psi'$. Consequently, there exists a scalar $a\in F^*$ and a
quadratic form $\eta$ such that $\psi'\cong a\psi \perp \eta$. The
form $\psi \perp a\psi$ is a subform of $\phi$ and is similar to
a $(k+1)$-fold Pfister form, contradicting the maximality of $k$.
Hence, $\phi$ is similar to a Pfister form.\vskip2mm \noindent{(2)
If $\phi$ is singular: In this case we need some results on the
isotropy problem, and the relation between the height and the type
of a singular form.

\begin{prop}(\cite[Proposition 1.1]{lagh1})
Let $F$ be a field of characteristic $2$, and let $\phi$, 
$\psi$ be anisotropic quadratic forms.\\ (1) If $\dim \phi=2$ and
$\dim \psi \geq 3$, then $\phi_{F(\psi)}$ is anisotropic.\\ (2) If
$\phi$ is nonsingular of dimension $2$ and $\psi$ is totally
singular, then $\phi_{F(\psi)}$ is anisotropic. \label{pia}
\end{prop}
\begin{prop} Let $\phi$ be as above. Then, $s=1$.
\label{paa}
\end{prop}

\prf Assume $s\geq 2$. Since $\h (\phi) \leq 1$, it follows that
$$\phi_{F(\phi)} \cong r\times \H \perp (s-1)\times \left[
0\right] \perp \left[ \alpha\right]$$ for some $\alpha \in
F(\phi)$. Lemma \ref{luf} implies that the $F(\phi)^2$-vector
space spanned by $c_1, \cdots, c_s$ is of dimension $\leq 1$. In
particular, the $F(\phi)^2$-vector space spanned by $c_1, c_2$  is
of dimension $\leq 1$, which means that the form $\left[
c_1\right] \perp \left[ c_2\right]$ becomes isotropic over
$F(\phi)$, a contradiction with Proposition \ref{pia}.\cn We
consider the following quadratic forms $$\mu:= [a_1, b_1] \perp
\cdots \perp [a_r, b_r],$$ and $$\nu:= [a_1, b_1] \perp \cdots
\perp [a_r, b_r] \perp c_1\left[ 1, \tr (\mu)\right].$$

\begin{lemma} Let $\mu$ and $\nu$ be as above. Then,
$\nu$ is anisotropic and $\nu_{F(\phi)}$ is hyperbolic.
\label{laa}
\end{lemma}
\prf Since $\phi_{F(\phi)} \prec \nu_{F(\phi)}$ and $\phi_{F(\phi)}$ 
contains $r$ hyperbolic planes, it follows
that $\nu_{F(\phi)} \cong r\times \H \perp \xi$ for some
$2$-dimensional quadratic form $\xi$. By comparing Arf invariant
we get $\xi \cong \H$. Therefore, $\nu_{F(\phi)}$ is hyperbolic.
The quadratic form $\nu$ is not hyperbolic because $\phi$ is
anisotropic. Assume that $\nu$ is isotropic. Thus, $2\leq \dim
\nu_{an} \leq \dim \nu -2 <\dim \phi$ and
$(\nu_{an})_{F(\phi)}\sim 0$, a contradiction with Proposition
\ref{lpp}.\cn\vskip1.5mm To complete the proof, let $k$ be the
maximal integer such that there exists a $k$-fold Pfister form
$\psi$ as a subform of $\nu$. We have to prove $\dim \nu=2^{k+1}$.
Assume $\dim \nu > 2^{k+1}$, and let $\psi'$ be a quadratic form
such that $\nu \cong \psi \perp \psi'$. Notice that $\dim \psi'
\leq \dim \nu -2<\dim \phi$. Since $\psi$ is anisotropic, $\phi$
is not totally singular and $c\neq 0$, it follows that
$F(\psi)(\phi)$ is well defined. However, $\nu_{F(\psi)(\phi)}
\sim \psi'_{F(\psi)(\phi)}\sim 0$. By Proposition \ref{lpp}, we
have $\psi'_{F(\psi)} \sim 0$, and thus $\psi$ is weakly dominated
by $\psi'$. Hence, there exists $a\in F^*$ and a quadratic form
$\eta$ such that $\psi' \cong a\psi \perp \eta$. The form $\psi
\perp a\psi$ is a subform of $\nu$ and is similar to a
$(k+1)$-fold Pfister form, contradicting the maximality of $k$.
Hence, $\nu$ is similar to a Pfister form.\cn

We recall a general result on the isotropy problem.

\begin{prop}(\cite[Corollaire 3.3]{lagh1})
Let $F$ be a field of characteristic $2$, and let $\phi$, $\psi$
be anisotropic quadratic forms over $F$ of dimension $\geq 2$ such
that $\phi$ is totally singular and $\psi$ is not totally
singular. Then, $\phi_{F(\psi)}$ is anisotropic. \label{piu}
\end{prop}

As a corollary we have:

\begin{cor} Let $F$ be a field of characteristic $2$, and let
$\phi$, $\psi$ be anisotropic quadratic forms. Assume that $\psi$
is not totally singular and $\phi$ is totally singular. Let $(F_i,
\phi_i)_{0\leq i\leq \h (\phi)}$ be the generic splitting tower of
$\phi$. Then, $F_i(\psi)$ is well defined for all $i\in \{0,
\cdots, \h(\phi)\}$. \label{ceaa}
\end{cor}

\prf Set $\psi=\left[ a_1, b_1\right] \perp \cdots \perp \left[
a_r, b_r \right] \perp \left[c_1\right] \perp \cdots \perp
\left[c_s\right]$. By assumption we have $r\geq 1$. Let $i\in\{ 0,
\cdots, \h (\phi)\}$ and assume that $F_i(\psi)$ is not defined.
Hence,
\begin{equation}
\psi_{F_i}\cong \H \perp \left[0\right] \perp \cdots \perp
\left[0\right] \label{eque} \end{equation} In particular, $r=1$.
If $s\geq 1$, then the $F_i^2$-vector space spanned by $\{c_1,
\cdots, c_s\}$ is of dimension $\geq 1$, a contradiction with
Lemma \ref{luf} and equation (\ref{eque}). Hence, $\psi=\left[a_1,
b_1\right]$ and $\psi_{F_i}\cong \H$. Since, $\phi_j$ is totally
singular for all $j\in \{0, \cdots, \h(\phi)\}$, it follows from
Proposition \ref{pia} (2) that $\psi_{F_i}$ is anisotropic, a
contradiction.\cn \vskip2mm


In the following theorem, we give some general results on the
generic splitting tower of a quadratic form.

\begin{theorem}
Let $\phi$ be an anisotropic quadratic form of height $h=\h
(\phi)$ and type $(r, s)$, and let $(F_i, \phi_i)_{0\leq i \leq
\h(\phi)}$ be its generic splitting tower. When $s\geq 1$, we
denote by $\eta$ the quasilinear part of $\phi$. Then, we have the
following statements:\\ (1) If $\phi$ is totally singular and
$\psi$ is not a totally singular quadratic form, then $\h (\phi)=
\h (\phi_{F(\psi)})$.\\ (2) If $s \geq 2$, then there exists $j\in
\{ 0, \cdots ,\h (\phi)-1\}$ such that $\phi_j$ is of type $(0,
s)$. More precisely, we have $\phi_{j}\cong \eta_{F_j}$.\\ (3) If
$r , s\geq 1$, then $\h (\eta) +1\leq \h (\phi) \leq \h(\eta) +
r$.\\ (4) Assume that $r, s\geq 1$ and there exists $j\in \{ 0,
\cdots , \h (\phi)-1\}$ such that $\phi_j$ is of type $(r_j , 1)$.
Then $s=1$.\\ (5) Let $j\in \{ 0, \cdots ,\h (\phi)\}$. Assume
that $\phi_j$ is of type $(r_j, \epsilon)$ with $r_j \geq 1$.
Then, for all $i\in \{ 0, \cdots, j\}$ the type of $\phi_i$ is
$(r_i, \epsilon)$ for some integer $r_i$.\\ (6) If $s\geq 2$, then
$\phi_{\h (\phi) -1}$ is totally singular of dimension $2$.\\ (7)
If $s=1$, then for all $i\in \{ 0, \cdots ,\h (\phi)\}$ the type
of $\phi_i$ is $(r_i ,1)$ for some $r_i$.\\ (8) If $s=0$, then
$\phi_{\h (\phi)-1}$ is similar to a Pfister form.\cn \label{tr}
\end{theorem}
\vskip2mm

\prf (1) Set $\psi=\left[
a_1, b_1\right] \perp \cdots \perp \left[ a_r, b_r \right] \perp
\left[c_1\right] \perp \cdots \perp \left[c_s\right]$ and
$\mu=\left[a_1, b_1\right]$. By Corollary \ref{ceaa} the fields
$F_i(\psi)$, $F_i(\mu)$ are well defined.\\

(i) If $\mu=\psi$, then by Proposition \ref{pia} we deduce that
$\mu_{F_i}$ is anisotropic, and thus by Proposition \ref{piu}
$(\phi_i)_{F_i(\mu)}$ is anisotropic.\\

(ii) If $\dim \psi>\dim \mu$, then $\dim \psi \geq 3$. By the same
argument as in the proof of Corollary \ref{ceaa} and by using
$\dim \psi \geq 3$, we deduce that $F_i(\mu)(\psi)$ is well
defined. The extension $F_i(\mu)(\psi)/F_i(\mu)$ is purely
transcendental \cite[Lemma 1]{ta}. If $(\phi_i)_{F_i(\psi)}$ is
isotropic, then $(\phi_i)_{F_i(\mu)}$ is also isotropic, a
contradiction with Proposition \ref{piu}.\\ Since,
$(\phi_i)_{F_i(\psi)}$ is anisotropic for all $i\in \{0, \cdots,
\h (\phi)\}$, it follows that $\h(\phi)=\h (\phi_{F(\psi)})$.\\

(2) Since $\dim \phi \geq s\geq 2$, it follows that $\h (\phi)\geq
1$. Let $j$ be the smallest in $\{ 1, \cdots, \h(\phi)\}$ such
that $\phi_j$ is of type $(r_j, s_j)$ with $s_j <s$. By the choice
of $j$, the form $\eta_{F_{j-1}}$ is anisotropic. Lemma \ref{luf}
implies that $\eta_{F_j}$ is isotropic. Since
$F_j=F_{j-1}(\phi_{j-1})$, it follows from Proposition \ref{piu}
that $\phi_{j-1}$ is totally singular. Therefore, by the
minimality of $j$ we conclude that $\phi_{j-1}$ is of type $(0,
s)$, and thus by Lemma \ref{luf} we obtain $\phi_{j-1}\cong
\eta_{F_{j-1}}$.\\

(3) The statement is clear for $s=1$. So we may assume $s\geq 2$.
By statement (2), there exists $j\in\{ 0, \cdots ,\h (\phi) -1\}$
such that $\phi_j\cong \eta_{F_j}$. Since $\phi$ is not totally
singular and $\phi_j$ is totally singular, we obtain $j\geq
1$. Since $\phi_{j-1}$ is not totally singular, it follows that
$\phi_l$ is not totally singular for all $l\in \{ 0, \cdots,
j-1\}$. By statement (1) we conclude that $\h
(\eta_{F_j})=\h(\eta_{F_{j-1}})=\cdots =\h (\eta)$. Since, $\h
(\phi) \geq \h (\phi_{j-1})=\h (\phi_j) +1$ and $\eta_{F_j}\cong
\phi_j$, we conclude that $\h (\phi) \geq \h (\eta)+1$. The second
inequality is obvious.\\

(4) There is nothing to prove in the case $j=0$. So we may assume
$j\geq 1$. Let $(r_{j-1}, s_{j-1})$ be the type of $\phi_{j-1}$.
Since $\phi_j$ is of type $(r_j, 1)$ and $j\leq \h (\phi)-1$, it
follows that $r_j >0$, otherwise $\h (\phi_{j-1})=1$ and $j=\h
(\phi)$. In particular, $r_{j-1}>0$ and $\phi_{j-1}$ is not
totally singular. Set $\phi_{j-1} =\xi_{j-1} \perp \eta_{j-1}$
with $\xi_{j-1} \in W_q(F_{j-1})$ and $\eta_{j-1}$ a totally
singular form over $F_{j-1}$. If $s_{j-1}\geq 2$, then
$\eta_{j-1}$ becomes isotropic over $F_{j-1}(\phi_{j-1})$, a
contradiction with Proposition \ref{piu}. Hence $s_{j-1}=1$. We
finish the proof by a simple iteration argument.\\

(5) There is nothing to prove in the case $j=0$. So we may assume
$j\geq 1$. Then, the type $(r_{j-1}, s_{j-1})$ of $\phi_{j-1}$
satisfies $r_{j-1} \geq r_j\geq 1$ and $s_{j-1}\geq \epsilon$. In
particular, $\phi_{j-1}$ is not totally singular. Set
$\phi_{j-1}=\xi_{j-1} \perp \eta_{j-1}$ with $\xi_{j-1} \in
W_q(F_{j-1})$ and $\eta_{j-1}$ a totally singular form over
$F_{j-1}$. If $s_{j-1}> \epsilon$, then $\eta_{j-1}$ becomes
isotropic over $F_j=F_{j-1}(\phi_{j-1})$, a contradiction with
Proposition \ref{piu}. Hence, $s_{j-1}=\epsilon$. We finish the
proof by a simple iteration argument.\\

(6) Assume $s\geq 2$. By statement (2) there exists $j\in \{ 0,
\cdots, \h (\phi)-1\}$ such that $\phi_j$ is of type $(0, s)$. In
particular, $\phi_{\h(\phi)-1}$ is totally singular. It follows
from Theorem \ref{tp} that $\phi_{\h(\phi) -1}$ is totally
singular of dimension $2$.\\

(7) Obvious.\\

(8) If $s=0$. Then, all forms $\phi_i$ are nonsingular. By Theorem
\ref{tp} the form $\phi_{h-1}$ is similar to a Pfister form.\cn

The notion of a Pfister neighbor form was extended in
characteristic $2$ as follows.
\begin{definition} (\cite[D\'efinition 1.2]{lagh1})
A quadratic form $\phi$ is a Pfister
neighbor if there exists an $n$-fold Pfister form $\pi$ such that $\dim
\phi>2^n$
and $\phi$ is weakly dominated by $\pi$. We call $\dim \pi -\dim \phi$
the codimension of $\phi$.
\label{dv}
\end{definition}
In \cite[Proposition 3.1]{lagh1}, it is shown that an anisotropic
Pfister neighbor can't be totally singular, and if $\phi$ is a
Pfister neighbor of $\pi$ then $\pi$ is unique up to isometry.
\vskip3mm

\begin{sloppypar}
Theorem \ref{tp} allows us to extend the notion of degree of a
quadratic form. So, let $\phi$ be a non-split quadratic form of
height $h=\h (\phi)$, and let $(F_i, \phi_i)_{0\leq i\leq h}$ be
its generic splitting tower. Assume that $\dim \phi_0 \geq 2$.
Then, $\h (\phi) \geq 1$. The field $F_{h-1}$ is called the
leading field of $\phi$ \cite{knebusch1}. Since $\phi_{h-1}$ is of
height $1$, it follows from Theorem \ref{tp} that $\phi_{h-1}$ is
either a Pfister neighbor of codimension $0$ or $1$ (in the sense
of Definition \ref{dv}), or a singular form of dimension $2$. We
have two possibilities:\\

(1) If $\phi_0$ is nonsingular, then $\phi_{h-1}$ is also
nonsingular. Hence, $\phi_{h-1}\cong a\pi$ for some $a\in
F_{h-1}^*$ and $\pi \in P_dF_{h-1}$ for some integer $d$. In this
case, we say that $\phi$ is of degree $d+1$. The form $\pi$ is
called the leading form of $\phi$ and is uniquely determined by
$\phi$ (we use the same argument as in \cite{knebusch1}).\\

(2) If $\phi_0$ is singular, then Theorem \ref{tr} implies that
$\phi_{h-1}$ is also singular. In this case, we say that $\phi$ is
of degree $0$. By Theorem \ref{tp} $\phi_{h-1}$ is of type
$(r_{h-1}, 1)$ with $r_{h-1}\geq 1$ or of type $(0, 2)$. When the
type is $(r_{h-1}, 1)$, we conclude by Theorem \ref{tp} that
$\phi_{h-1}\cong \xi \perp \left[ c\right]$ with $\xi \in
W_q(F_{h-1})$ and $c \in F_{h-1}^*$ such that $c\xi \perp \left[
1, \tr (\xi)\right]$ is a Pfister form that we call also the
leading form of $\phi$ \cite{knebusch1}.\\
\end{sloppypar}

If $\phi$ is a split form (resp. $\dim \phi_{an}=1$), we set the
degree of $\phi$ as $\infty$ (resp. we set the degree of $\phi$ as
$0$).\\

We denote by $\deg (\phi)$ the degree of $\phi$, $J_n(F)$ the set
of all quadratic forms of degree at least $n$. Clearly, $GP_nF
\subset J_{n+1}(F)$ and $W_q(F)=J_1(F)$.\\

As in \cite{knebusch1}, we have the following proposition.
\begin{prop}
(1) Let $\tau \in P_nF$ anisotropic and $a\in F^*$. Let $\phi$ be
a quadratic form of degree $\geq n+2$. Then, the quadratic form
$a\tau \perp \phi$ is of degree $n+1$.\\ (2) For $n\geq 0$, the
set $J_n(F)$ is closed under addition. In particular, $J_n(F)$ is
a subgroup of $W_q(F)$ for $n\geq 1$, and $I^nW_q(F) \subset
J_{n+1}(F)$. \label{lde}
\end{prop}
\prf We use the same proof as in \cite[Theorem 6.3 and
6.4]{knebusch1}.\cn The following proposition allows us to define
the complementary form of a Pfister neighbor in characteristic
$2$.
\begin{prop}
Let $\xi \in W_q(F)$ and $c_1, \cdots, c_s\in F$ such that
$$\phi:=\xi \perp \left[c_1\right] \perp \cdots \perp
\left[c_s\right]$$ is a Pfister neighbor. Let $\delta \in W_q(F)$
and $d_1, \cdots, d_s\in F$ such that $$\xi \perp \left[c_1,
d_1\right] \perp \cdots \perp \left[c_s, d_s\right] \perp \delta$$
is similar to a Pfister form.\\ (1) If $F(\phi)$ is well defined,
then $\phi_{F(\phi)} \sim (\left[c_1\right] \perp \cdots \perp
\left[c_s\right] \perp \delta)_{F(\phi)}$.\\ (2) The quadratic
form $\left[c_1\right] \perp \cdots \perp \left[c_s\right] \perp
\delta$  is unique up to isometry. \label{pva}
\end{prop}

\prf Assume that $1\in D_F(\phi)$.\\ (1) The form $\pi:=\xi \perp
\left[c_1, d_1\right] \perp \cdots \perp \left[c_s, d_s\right]
\perp \delta$ is a Pfister form. Since $\pi_{F(\phi)}$ is
isotropic, it follows that $$\xi_{F(\phi)} \sim (\left[c_1,
d_1\right] \perp \cdots \perp \left[c_s, d_s\right] \perp
\delta)_{F(\phi)}.$$ In particular, $$\phi_{F(\phi)} \sim
(\left[c_1\right] \perp \cdots \perp \left[c_s\right] \perp
\left[c_1, d_1\right] \perp \cdots \perp \left[c_s, d_s\right]
\perp \delta)_{F(\phi)}.$$ By Lemma \ref{l1} we have $\left[ c_i,
d_i\right] \perp \left[ c_i\right] \sim \left[ c_i\right]$. Hence,
$$\phi_{F(\phi)} \sim (\left[c_1\right] \perp \cdots \perp
\left[c_s\right] \perp \delta)_{F(\phi)}.$$(2) Let $d'_1, \cdots,
d'_s\in F$ and $\delta' \in W_q(F)$ such that $\pi':=\xi \perp
\left[ c_1, d'_1\right] \perp \cdots \left[c_s, d'_s\right] \perp
\delta'$ is another Pfister form. However, $\phi$ is a Pfister
neighbor of $\pi$ and $\pi'$. By \cite[Proposition 3.1]{lagh1}
$\pi \cong \pi'$. In particular, $$\pi \perp \left[c_1 \right]
\perp \cdots \perp \left[ c_s\right] \cong \pi' \perp \left[
c_1\right] \perp \cdots \perp \left[ c_s\right].$$By Lemma
\ref{l1} and the Witt cancellation (Proposition \ref{gwc}) we get
the desired conclusion.\cn
\begin{sloppypar}
\begin{definition}
With the same notations and hypothesis as in Proposition
\ref{pva}, the quadratic form $\left[c_1\right] \perp \cdots \perp
\left[c_s\right] \perp \delta$ is called the partial complementary
form of $\phi$.
\end{definition}
Obviously if $\phi$ is a Pfister neighbor of type $(r, s)$, then
the partial complementary form $\phi'$ of $\phi$ is of type $(r',
s)$ for some integer $r'$. In characteristic not $2$, $\phi'$ is
known as the complementary form of $\phi$.\\

In characteristic $2$, we extend the notion of excellent form as
follows.

\begin{definition}
Any form of dimension $\leq 1$ is called excellent. A quadratic
form of dimension $\geq 2$ is called excellent if it is a Pfister
neighbor and its partial complementary form is excellent.
\label{dfe}
\end{definition}
\end{sloppypar}
In collaboration with Mammone \cite{ap} we extended in
characteristic $2$ a theorem of Hoffmann on the isotropy of
quadratic forms \cite{h}. Here is our result.
\begin{theorem} (\cite{ap}) Let $F$ be a field of characteristic $2$, and
let $\phi$, $\psi$ be anisotropic quadratic forms over $F$. Assume
that:\\ (1) If $\phi$ is nonsingular, then $\dim \phi \leq 2^n <
\dim \psi$ for some integer $n\geq 1$.\\ (2) If $\phi$ is singular
of type $(r, s)$, then $2r +2s \leq 2^n <\dim \psi$ for some
$n\geq 1$.\\ Then, $\phi$ remains anisotropic over $F(\psi)$.
\label{tis}
\end{theorem}
As a corollary of Theorem \ref{tis}, we have:
\begin{cor} Let $\phi$ be an anisotropic
Pfister neighbor of $\pi\in P_nF$ and $\psi$ be its partial
complementary form. Then, we have $(\phi_{F(\phi)})_{an}\cong
\psi_{F(\phi)}$ in the following cases:\\ (1) $\phi$ is
nonsingular.\\ (2) $\phi$ is singular of type $(r, s)$ such that
the type $(r', s)$ of $\psi$ satisfies $2r'+2s \leq 2^n$.\\ (3)
$\psi$ is a Pfister neighbor. \label{lis}
\end{cor}

\prf Let $(r, s)$ be the type of $\phi$ and $(r', s)$ be the type
of $\psi$. We have $\dim \psi=2r' +s$ and $\phi_{F(\phi)} \sim
\psi_{F(\phi)}$ (Proposition \ref{pva}). Since $\dim \phi>2^n$, it
follows that $\dim \psi <2^n$. The form $\phi_{F(\pi)}$ is
isotropic. Hence, $F(\pi)(\phi)/F(\pi)$ is a purely transcendental
extension \cite[Corollaire 3.4]{lagh1}.\\ $\bullet$ Assume that
$\phi$ satisfies the case (1) or (2). If $\psi_{F(\phi)}$ is
isotropic, then $\psi_{F(\pi)}$ is isotropic, a contradiction with
Theorem \ref{tis}. Hence $\psi_{F(\phi)}$ is anisotropic.\\
$\bullet$ Assume that $\psi$ is a Pfister neighbor of a $k$-fold
Pfister form $\rho$. Since $\dim \psi >2^k$, we obtain $k<n$, that
means, $\dim \rho <\dim \pi$. If $\psi_{F(\phi)}$ is isotropic,
then $\psi_{F(\pi)}$ is isotropic. Consequently, $\rho_{F(\pi)}$
is isotropic and thus hyperbolic, a contradiction with Proposition
\ref{lpp}.\cn

\begin{definition}
For a field extension $K/F$, we say that a quadratic form $\phi$
over $K$ is defined over $F$ if there exists a quadratic form
$\psi$ over $F$ such that $\phi \cong \psi_K$.
\end{definition}


For an excellent quadratic form, we prove a general result which
generalizes a known result of Knebusch on such a form in
characteristic not $2$, and we prove that an anisotropic singular
excellent form has a particular type.
\begin{prop} Let $\phi$ be an anisotropic excellent form of
dimension $\geq 3$, and let $(F_i, \phi_i)_{0\leq i \leq \h
(\phi)}$ be its generic splitting tower.\\ (1) If $\phi$ is
singular, then $\phi$ is of type $(r, 1)$ for some integer $r>0$.
In particular, all quadratic forms $\phi_i$ are of type $(r_i,
1)$.\\ (2) All quadratic forms $\phi_i$ are defined over $F$.
\label{pe}
\end{prop}
\begin{sloppypar}
\prf (1) Let $\psi$ be the partial complementary form of $\phi$.
Since $\phi$ is a Pfister neighbor, it follows from
\cite[Proposition 3.1]{lagh1} that $\phi$ is not totally singular.
We will proceed by induction on $\h (\phi)$. If $\h (\phi)=1$,
then the result is an immediate consequence of Theorem \ref{tp}.
Now assume that the result is true for any anisotropic excellent
singular form of height $<\h (\phi)$. By Corollary \ref{lis} (3)
$(\phi_{F(\phi)})_{an}\cong \psi_{F(\phi)}$. Hence,
$\psi_{F(\phi)}$ is excellent of height $\h (\phi)-1$. Since
$\psi_{F(\phi)}$ is singular, we get by the induction hypothesis
that $\psi_{F(\phi)}$ is of type $(r', 1)$ for some integer $r'$.
If $r'=0$, then $\h(\phi)=1$ and the result is true by Theorem
\ref{tp}. If $r'>0$, then by Theorem \ref{tr} (5) $\phi$ is of
type $(r, 1)$ for some integer $r>0$.\\ (2) A consequence of
Proposition \ref{pva} and Corollary \ref{lis} (3).\cn
\end{sloppypar}
Now we give some partial results on quadratic form of height $2$
and degree $1$ or $2$.

\begin{theorem}
Let $F$ be a field of characteristic $2$, and let $\phi$ be an
anisotropic nonsingular quadratic form over $F$.\\ (1) $\phi$ is
of height $2$ and degree $1$ if and only if $\phi$ is one of the
following types:\\ $\bullet$ $\phi$ is excellent of codimension
$2$;\\ $\bullet$ $\phi$ is of dimension $4$ with non trivial Arf
invariant.\\ (2) $\phi$ is excellent of height and degree $2$ if
and only if $\phi_{F(\phi)}$ is not hyperbolic and
$(\phi_{F(\phi)})_{an} \in GP_1F(\phi)$ is defined over $F$.\\ (3)
$\phi$ is of height and degree $2$ whose leading form is not
defined over $F$ if and only if $\phi$ is an Albert form, i.e. a
nonsingular quadratic form of dimension $6$ with trivial Arf
invariant. \label{thd}
\end{theorem}

The characterization of anisotropic quadratic forms of height
$2$ and degree $1$ or $2$ is complete in characteristic not $2$. Here, we
can't
settle the case when $\phi$ is of height
and degree $2$ such that $(\phi_{F(\phi)})_{an}$ is not defined over $F$
and
the leading form of $\phi$ is defined over $F$.

\begin{prop}
Let $F$ be a field of characteristic $2$, and let $\phi$ be an
anisotropic nonsingular quadratic form of dimension $4$ with non
trivial Arf invariant, or an Albert form. We have the following
statements:\\ (1) If $\dim \phi=4$, then $\phi$ is of height $2$,
degree $1$ and $(\phi_{F(\phi)})_{an}$ is not defined over $F$.\\
(2) If $\phi$ is an Albert form, then $\phi$ is of height and
degree $2$ with leading form not defined over $F$. In particular,
$(\phi_{F(\phi)})_{an}$ is not defined over $F$. \label{lhy}
\end{prop}

\prf (1) Since $\dim \phi=4$ and $\tr (\phi)\neq 0$, we have $\phi
\not\in GP_1F$ and $\dim (\phi_{F(\phi)})_{an}=2$. Hence $\h
(\phi)=2$ and $\deg (\phi)=1$. Assume that $(\phi_{F(\phi)})_{an}$
is defined over $F$ and let $\alpha \in F^*$ such that
$\phi_{F(\phi)} \sim (\alpha \left[ 1, \tri
(\phi)\right])_{F(\phi)}$. Hence, $\mu:=\phi \perp \alpha \left[
1, \tri (\phi)\right]$ is hyperbolic over $F(\phi)$. Let $\rho$ be
a $3$-dimensional quadratic form dominated by $\phi$. Clearly,
$\rho$ is a Pfister neighbor of some $1$-fold Pfister form $\tau
\in P_1F$. The forms $\phi_{F(\rho)}$ and $\rho_{F(\tau)}$ are
isotropic, thus the extensions $F(\rho)(\phi)/F(\rho)$ and
$F(\tau)(\rho)/F(\tau)$ are purely transcendentals
\cite[Corollaire 3.4]{lagh1}. Consequently, $\mu_{F(\tau)} \sim
0$. The form $\mu$ is not hyperbolic because $\phi$ is
anisotropic. By Corollary \ref{caa} we deduce that $\mu$ is
isotropic and $\mu_{an}$ is similar to $\tau$. Therefore,
$\tau_{F(\phi)}\sim 0$. By Proposition \ref{lpp} $\phi$ is similar
to $\tau$, a contradiction.\\ (2) Assume that $\phi$ is an Albert
form. By Theorem \ref{tp}, we have $\h (\phi) \geq 2$. Since $\tri
(\phi)=0$, we conclude that $(\phi_{F(\phi)})_{an} \in
GP_1F(\phi)$. Hence, $\h (\phi)=2$. Let $\tau \in P_1F(\phi)$ be
the leading form of $\phi$. Assume that $\tau$ is defined over
$F$. Consequently, $\phi_{F(\tau)(\phi)} \sim 0$. If
$\phi_{F(\tau)}$ is anisotropic, then $\phi_{F(\tau)}$ is of
height $1$, a contradiction with Theorem \ref{tp}. Hence
$\phi_{F(\tau)}$ is isotropic and thus $F(\tau)(\phi)/F(\tau)$ is
a purely transcendental extension \cite[Corollaire 3.4]{lagh1}. We
conclude that $\phi_{F(\tau)} \sim 0$ and by Corollary \ref{caa}
$\phi$ is isotropic, a contradiction.\cn
\begin{sloppypar}
\begin{prop}
Let $F$ be a field of characteristic $2$, and let $\phi, \eta \in
W_q(F)$ be anisotropic forms such that $(\phi_{F(\phi)})_{an}\cong
\eta_{F(\phi)}$ and $\dim \eta =2$ or $\eta \in GP_1F$. Then,
$\phi$ is an excellent form of partial complementary form $\eta$.
\label{phd}
\end{prop}
\end{sloppypar}
\begin{sloppypar} \prf Set $\tri =\tri (\phi)$ and $\mu =\phi
\perp \eta$. We have $2\dim \phi >\dim \mu$. The form $\mu$ is not
hyperbolic because $\phi$ is anisotropic, and we have
$\mu_{F(\phi)} \sim 0$. We have to prove that $\mu$ is similar to
an anisotropic Pfister form.\\

\noindent{\bf (1)} If $\dim \eta=2$: In this case $\tri \neq 0$.
Let $\alpha \in F^*$ such that $\eta=\alpha \left[ 1, \tri
\right]$.

(1.1) Assume that $\mu$ is isotropic. Hence, $$\phi \cong e\left[
1, f\right] \perp \phi'$$ and $$\alpha \left[ 1, \tri \right]
\cong e\left[ 1, \tri \right]$$ for some quadratic form $\phi'$
and $e, f\in F^*$. It follows that
\begin{equation}
(\phi' \perp e\left[ 1,  f+ \tri \right])_{F(\phi)}\sim 0
\label{ehy}
\end{equation}
The forms $\phi' \perp e\left[ 1,  f+ \tri \right]$ and $\phi$
have the same dimension.

$\bullet$ If $\phi' \perp e\left[ 1,  f+ \tri \right]$ is
isotropic, it follows from Proposition \ref{lpp} and equation
(\ref{ehy}) that $\phi'\perp  e\left[ 1,  f+ \tri \right]\sim 0$.
Since $\phi'$ is anisotropic, we deduce $\phi'\cong  e\left[ 1, f+
\tri \right]$. Hence $\phi$ is isotropic, a contradiction.

$\bullet$ If $\phi' \perp e\left[ 1,  f+ \tri \right]$ is
anisotropic, then Proposition \ref{lpp} implies that $\phi$ and
$\phi'\perp e\left[ 1,  f+ \tri \right]$ are similar. In
particular, $\phi_{F(\phi)}\sim 0$, a contradiction.

Hence, $\mu$ is anisotropic.

(1.2) Assume that $\mu_{F(\mu)}$ is not hyperbolic and let
$\mu_1=(\mu_{F(\mu)})_{an}$. As in case (1.1) we have 
$\phi_{F(\mu)} \cong \beta \mu_1$ for
some $\beta \in F(\mu)^*$, and thus $\dim \phi< \dim \mu$ and
$\phi_{F(\phi)(\mu)}\sim 0$. Therefore by Proposition \ref{lpp}
$\phi_{F(\phi)} \sim 0$, a contradiction.

Hence, $\mu_{F(\mu)}$ is hyperbolic, and thus $\mu$ is similar to
a Pfister form.\\

\noindent{{\bf (2)}} If $\dim \eta=4$: We may assume $\eta
\in P_1F$. Set $\eta =\left[ 1, y\right] \perp z\left[ 1,
y\right]$.

(2.1) If $\mu$ is isotropic, then $$\phi \cong r\left[ 1, x\right]
\perp \phi''$$ and $$\eta \cong r\left[ 1, y\right] \perp rz\left[
1, y\right],$$ for some quadratic form $\phi''$ and $r\in F^*$. In
particular, $$(\phi'' \perp r\left[ 1, x+y \right] \perp rz\left[
1, y\right])_{F(\phi)} \sim 0.$$ Since $\phi$ is anisotropic, it
follows that $\phi'' \perp r\left[ 1, x+y \right] \perp rz\left[
1, y\right]$ is not hyperbolic.

(i) If $\phi'' \perp r\left[ 1, x+y \right] \perp rz\left[ 1,
y\right]$ is anisotropic, then Proposition \ref{lpp} implies that
$$\phi'' \perp r\left[ 1, x+y \right] \perp rz\left[ 1, y\right]
\cong u\phi \perp \xi$$ for some $2$-dimensional form $\xi$ and
$u\in F^*$. By comparing Arf invariant we deduce that $\xi \cong
\H$, a contradiction.

(ii) If $\phi'' \perp r\left[ 1, x+y \right] \perp rz\left[ 1,
y\right]$ is isotropic, then $\dim (\phi'' \perp r\left[ 1, x+y
\right] \perp rz\left[ 1, y\right])_{an} \leq \dim \phi$. By
Proposition \ref{lpp} we get that $\phi$ is similar to $(\phi''
\perp r\left[ 1, x+y \right] \perp rz\left[ 1, y\right])_{an}$. In
particular, $\phi_{F(\phi)} \sim 0$, a contradiction.

Hence, $\mu$ is anisotropic.

(2.2) If $\phi_{F(\mu)}$ is isotropic, then $F(\mu)(\phi)/F(\mu)$
is a purely transcendental extension \cite[Corollaire 3.4]{lagh1}.
Hence, $\mu_{F(\mu)}\sim 0$ and thus $\mu$ is similar to a Pfister
form.

(2.3) If $\mu_{F(\mu)}$ is not hyperbolic, then by case (2.2)
$\phi_{F(\mu)}$ is anisotropic. Let $\mu_1=(\mu_{F(\mu)})_{an}$.
We have $(\mu_1)_{F(\mu)(\phi)} \sim 0$ and $\dim \mu_1 \leq \dim
\phi +2$. By the same arguments as in cases (i) and (ii) we obtain
that $\phi_{F(\mu)}$ is similar to $\mu_1$. In particular, $\dim
\phi<\dim \mu$ and $\phi_{F(\phi)(\mu)}\sim 0$. It follows from
Proposition \ref{lpp} that $\phi_{F(\phi)}\sim 0$, a
contradiction.

Hence, $\mu_{F(\mu)}$ is hyperbolic and thus $\mu$ is similar a
Pfister form.\cn
\end{sloppypar}
\noindent{{\it Proof of Theorem \ref{thd}.} Let $\phi$ be an
anisotropic nonsingular quadratic form. Set $\tri =\tri (\phi)$.\\

(1) If $\phi$ is excellent of codimension $2$, it follows from
Corollary \ref{lis} (3) that $\h (\phi)=2$ and degree $1$. By
Proposition \ref{lhy}, an anisotropic quadratic form of dimension
$4$ with non trivial Arf invariant is of height $2$ and degree
$1$. For the converse, assume that $\phi$ is of height $2$ and
degree $1$. In particular, $\dim \phi\geq 4$ and $\dim
(\phi_{F(\phi)})_{an}=2$. Let $\alpha \in F(\phi)^*$ such that
$\phi_{F(\phi)} \sim \alpha \left[ 1, \tri \right]$.\\

(1.1) If $\alpha \left[ 1, \tri \right]$ is defined over $F$, then
we get by Proposition \ref{phd} that $\phi$ is excellent of
codimension $2$.\\

(1.2) If $\alpha \left[ 1, \tri \right]$ is not defined over $F$.
We have $$C(\phi)_{F(\phi)} =\left[ \tri , \alpha \right) \in \br
(F(\phi)).$$ Assume that $\dim \phi \geq 6$. By Proposition
\ref{lhy} (2) $\phi$ is not an Albert form. However, by the
reduction index theorem \cite[Theorem 4]{ta} the Schur index of
the algebra $C(\phi)$ is less or equal to $2$. Let $\tau=\left[ 1,
r\right] \perp s\left[ 1, r\right]\in P_1F$ such that $C(\phi)
=C(\tau) \in \br (F)$. Clearly $$(\left[ 1, r\right] \perp s\left[
1, r\right])_{F(\phi)} \cong \left[ 1, \tri \right] \perp \alpha
\left[ 1, \tri \right].$$ Thus $(\left[ 1, r+\tri \right] \perp
s\left[ 1, r\right])_{F(\phi)} \sim \alpha \left[ 1, \tri
\right]$. By \cite[Th\'eor\`eme 1.3]{lagh1} the form $\left[ 1,
r+\tri \right] \perp s\left[ 1, r\right]$ is isotropic.
Consequently, $\alpha \left[ 1, \tri \right]$ is defined over $F$,
a contradiction. Hence, $\dim \phi=4$.\\

(2) A consequence of Proposition \ref{phd} and Corollary \ref{lis}
(3).\\

(3) The sufficient condition is a consequence of Proposition
\ref{lhy} (2). For the necessary condition, we follow the proof of
Kahn in characteristic not $2$ \cite{kahn} by using the reduction
index theorem in characteristic $2$ \cite{ta}.\cn

Let us finish this note with some questions. The first one
 is related to the case that we don't settle in Theorem \ref{thd}.

\begin{ques}
Let $\phi$ be an anisotropic quadratic form of height and degree
$2$ and let $\tau \in P_1F$ anisotropic such that
$(\phi_{F(\phi)})_{an}$ is similar to $\tau$ but not defined over
$F$. Is $\phi$ of dimension $8$? \label{q1}
\end{ques}

It is well known that Question \ref{q1} has a positive answer in
characteristic not $2$ \cite[1.6]{fitzgerald}. \vskip2mm

\begin{ques} Let $\phi$ be an anisotropic form, and let $(F_i,
\phi_i)_{0\leq i \leq \h (\phi)}$ be its generic splitting tower.
Assume that $\phi$ is of type $(r, \epsilon)$ such that
$\epsilon\leq 1$ and all forms $\phi_i$ are defined over $F$. Is
$\phi$ an excellent form? \label{q2}
\end{ques}

More precisely we ask the following question.

\begin{ques} Let $\phi$ be an anisotropic form. Assume that $\phi$ is
of type $(r,\epsilon)$ such that $\epsilon \leq 1$ and
$(\phi_{F(\phi)})_{an}$ is defined over $F$. Is $\phi$ a Pfister
neighbor? \label{q3}
\end{ques}
\begin{sloppypar}
Proposition \ref{phd} gives a positive answer to Question \ref{q2}
when $\dim (\phi_{F(\phi)})_{an}=2$ or $(\phi_{F(\phi)})_{an}\in
GP_1F(\phi)$.\\
\end{sloppypar}
To formulate Question \ref{q3} when $\phi$ is a singular form with
radical of dimension $\geq 2$, it is necessary to add other
hypothesis on the type of $\phi$. In fact, Proposition \ref{eg}
gives an example of an anisotropic singular quadratic form $\phi$
such that $(\phi_{F(\phi)})_{an}$ is defined over $F$ but $\phi$
is not a Pfister neighbor.
\begin{prop}
Let $F$ be a field of characteristic $2$, $n$ be an integer such
that $2+ 2n$ is not a power of $2$, and $X, Y, Z_1, \cdots , Z_n$
some variables over $F$. Let $\phi=\left[ X, Y\right]\perp \left[
Z_1\right] \perp \cdots \perp \left[ Z_n\right]$ and $K=F(X, Y,
Z_1, \cdots, Z_n)$. Then, $(\phi_{K(\phi)})_{an}$ is defined over $K$ but 
$\phi$ is not a Pfister neighbor. More
precisely, $(\phi_{K(\phi)})_{an}=(\left[ Z_1\right] \perp \cdots
\perp \left[ Z_n\right])_{K(\phi)}$. \label{eg}
\end{prop}

\prf Let $\eta=\left[ Z_1\right] \perp \cdots \perp \left[
Z_n\right]$. Assume that $\phi$ is a Pfister neighbor over $K$. A
simple computation implies that $2+2n$ is a power of $2$, a
contradiction. It follows from Proposition \ref{piu} that
$\eta_{K(\phi)}$ is anisotropic and thus $\phi_{K(\phi)}\cong \H
\perp \eta_{K(\phi)}$, as desired.\cn

\begin{thebibliography}{10}
\bibitem{b3} R. Baeza, \newblock{\it Ein Teilformensatz f\"ur quadratische
Formen in Charakteristik $2$,} \newblock{Math. Z. {\bf 135}
(1974), 175--184.}

\bibitem{b1} R. Baeza, \newblock{\it Quadratic forms over semilocal
rings,} \newblock{Lect. Notes Math. vol. {\bf 665}, Berlin Heidelberg New
York:
Springer 1978.}

\bibitem{b2} R. Baeza, \newblock{\it The norm theorem for quadratic forms
over a field of characteristic $2$,} \newblock{Comm. Algebra {\bf
(5) 18} (1990), 1337--1348.}

\bibitem{fitzgerald} R. W. Fitzgerald,
\newblock{\it Quadratic forms of height two}, \newblock{Trans. Amer. Math.
Soc. {\bf 283} (1984), 339--351.}

\bibitem{h} D. W. Hoffmann, \newblock {\it Isotropy of quadratic forms
over the function field of a quadric}, \newblock{Math. Z. {\bf
220} (1995), 461--476.}

\bibitem{kahn} B. Kahn, \newblock {\it Formes quadratiques de hauteur et
de
degr\'e 2,} \newblock{Indag. Math. {\bf 7} (1996), 47--66.}

\bibitem{kn1} M. Knebusch, \newblock{\it Specialization of quadratic
and symmetric bilinear forms, and a norm theorem,} \newblock{Acta
Arithmetica  {\bf XXIV} (1973), 279-- 299.}

\bibitem{knebusch1} M. Knebusch, \newblock {\it Generic splitting of
quadratic forms I}, \newblock{Proc. London. Math. Soc. {\bf 33}
(1976), 65--93.}

\bibitem{knebusch2} M. Knebusch, \newblock {\it Generic splitting of
quadratic
forms II,} \newblock{Proc. London Math. Soc. {\bf 34} (1977),
1--31.}

\bibitem{knebusch3} M. Knebusch, \newblock {\it Specialization of
quadratic forms in
characteristic $2$,} \newblock{Talk in the Conference
``Quadratische und Hermitische Formen'',} Oberwolfach, 1999.

\bibitem{lagh1} A. Laghribi, \newblock{\it Certaines formes quadratiques
de dimension
$\leq 6$ et corps des fonctions en caract\'eristique $2$,}
\newblock{to appear in Israel J. Math.}

\bibitem{ap} A. Laghribi, P. Mammone, \newblock{\it Isotropie d'une forme
quadratique
sur le corps des fonctions d'une quadrique projective en caract\'eristique
$2$,}
\newblock{Preprint 1999.}

\bibitem{ta} P. Mammone, J.-P. Tignol, A. Wadsworth, \newblock{\it Fields
of characteristic $2$ with prescribed $u$-invariant,}
\newblock{Math. Ann.  {\bf 290} (1991), 109--128.}

\bibitem{merkurjev} A. S. Merkurjev, \newblock {\it Alg\`ebres simples et
formes quadratiques,}
\newblock{ In Russian: {Izv. Akad. Nauk SSSR,} {\bf 55} (1991), 218--224.}
\newblock{ English translation: Math. USSR Izv., {\bf 38} (1992),
215--221.}

\bibitem{s} W. Scharlau, \newblock{\it Quadratic and Hemitian forms,}
\newblock{(Grundlehren Math. Wiss. Bd. 270) Berlin, Heidelberg, New York,
Tokyo:
Springer 1985.}

\bibitem{wadsworth} A. R. Wadsworth, \newblock {\it Noetherian pairs and
function fields of quadratic forms,} \newblock{Doctoral Dissertation, 
University of California, 1972.}
\end{thebibliography}
\end{document}


