\titlea{\S24}{ Some Open Problems in Characteristic $2$}
\leftrunningtitle{Chapter III. Some Applications}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S24  Some Open Problems in Characteristic $2$}

We subdivide this section into parts (A) -- (D), posing four problems which seem to be open.
\medskip

As before, the word ``form'' stands for nondegenerate quadratic form and the ground field $k$ is arbitrary, unless indicated otherwise.
\bigskip

\noindent {\bf (A)} Let $\Char k=2$. {\it Which nonregular forms of height~$1$ are there?}
\medskip

If $\varphi$ is anisotropic and  $\dim \varphi = 2 + \dim QL (\varphi)$, then $\varphi$ surely has height~$1$. Not every such form is excellent! As a simple example we take an anisotropic $4$-dimensional form 
$$\varphi  =  [a_1, a_2] \; \perp \; \left[ {a_3 \atop 1} \; {1 \atop b_3}
\right].$$
If $\varphi$ were excellent, it would have to be a Pfister form up to a scalar factor, which is not the case.

We give another example. Let 
$$\varphi  =  [a_1, a_2, a_3, a_4] \; \perp \; \left[ {a_5 \atop 1} \; {1 \atop
b_5} \right]$$
be anisotropic. $\{$E.g., let  $a_1,\dots, a_5, b_5$ be unknowns over a field $\kappa$ and let 
$k = \kappa (a_1, \dots, a_5, b_5)$.$\}$ We have $h(\varphi) = 1$ and $\dim \varphi = 6$.
Let  $\varphi < \tau$ and $\tau$ a strictly regular hull of $\varphi$ (cf. \S16). Let $\chi \colon =
QL(\varphi)$. By the Scholium in \S16 we have 
$$\varphi \; = \; \chi \perp \gamma, \quad \tau \; = \; \mu \perp \gamma$$
with $\chi < \mu$, $\dim \mu = 2 \dim \chi = 8$. Thus,  $\dim \tau = 8+2 = 10$. The form $\vf$ is certainly not a Pfister neighbour.

Let us ask a more precise question: {\it Do there exist any anisotropic forms of height~$1$, other than close Pfister neighbours and anisotropic forms $\varphi$ with  $\dim (\varphi) = 2 + \dim QL(\varphi)$?}
\bigskip

\noindent {\bf (B)} Let $\varphi$ be a neighbour of a Pfister form $\tau$ and let $\eta$ be the complementary form. Then $\varphi \otimes k(\varphi) \sim (-\eta) \otimes k(\varphi)$ (\S 20, Theorem~2). {\it Is $\eta \otimes k(\varphi)$ anisotropic?}
\medskip

If the characteristic of the ground field is different from two, this is true, as shown originally by R.~Fitzgerald [F]. Hence $(- \eta)\otimes k(\varphi)$ is the first higher kernel form of $\varphi$. 
Later on, D.~Hoffmann proved the following far-reaching theorem:

\begtheorem{1} {\rm [H$_4$]}. Let $\Char k\ne 2$ and let $\varphi$ and $\eta$ be anisotropic forms over $k$. Assume that there exists a $2$-power $2^m$ such that $\dim
\eta \leq 2^m < \dim \varphi$.  Then  $\eta \otimes k(\varphi)$ is anisotropic.
\endtheorem
%\medskip

\noi{\it Does this statement remain true in characteristic $2$?\/}\fonote{See Addendum at the end of this section!}
\medskip

One could simply try to extend the proofs of Fitzgerald and Hoffmann to characteristic two. This approach, however, leads to problems very soon since these proofs make extensive use of the following argument: let   $\psi_1$ and $\psi_2$ be anisotropic forms over $k$ whose orthogonal sum  $\psi_1 \perp\psi_2$ is isotropic. Then there exists an  $a \in k^*$ such that
$\psi_1 = [a] \perp \psi'_1$ and $\psi_2 = [-a] \perp\psi'_2$. But this is only correct when $k$ has characteristic $\ne2$.\fonote{In characteristic $2$ the argument remains valid only for {\it symmetric bilinear forms.}} In characteristic~$2$ there does not seem to exist anything that achieves  results similar to this simple, but effective argument. 
\bigskip

\noi {\bf (C) Definition.} Let $L/k$ be a field extension and $\psi$ a form over $L$. We say that $\psi$ {\it is definable over}\Idx{definable over a field} 
$k$ if there exists a form $\eta$ over $k$ such that $\psi
\cong \eta \otimes L$.
\medskip

Let $\varphi$ be a form over $k$. Let $(K_r \; \vert \; 0 \leq r \leq h)$ be a generic splitting tower of $\varphi$ and let  $\varphi_r \colon = \ker (\varphi \otimes K_r)$, the $r$-th higher kernel form of $\varphi$ $(0 \leq r \leq h)$. We are interested in the situation where all $\varphi_r$ are definable over $k$. One should observe that this property is independent of the choice
of the generic splitting tower  $(K_r \;\vert \; 0 \leq r \leq h)$.

\begproblem{} {\it For which anisotropic forms $\varphi$ over $k$ are all higher kernel forms of $\varphi$ definable over $k$?}
\endproblem

Theorem 3 from \S 22 tells us that this is the case for excellent forms. For $\Char k\ne 2$ it is known that there are no other forms with this property [K$_6$, Th.7.14]. $\{$For this reason many texts, e.g., also  [KS, \S 13], {\it define} excellence  via this property when  $\Char k\ne 2$. We will {\it not} do that  here.$\}$

Obviously every anisotropic form of height $1$ satisfies this property. Among such forms there are non-excellent ones, as seen under (A). In case the first problem under (B) has an affirmative answer,$^{35}$
%\fonote{See Addendum at the end of this section!}
 it is not so hard to obtain  further examples of anisotropic forms whose higher kernel forms are definable over the ground field. Simply embed $\eta$ into an anisotropic Pfister form $\tau$, of dimension $> 2 \dim \eta$ (we may assume without loss of generality that 
$1 \in D(\eta)$). Then all higher kernel forms of the polar $\varphi$ of $\eta$ in $\tau$ are also definable over $k$.
\bigskip

\noi {\bf (D)} Let $(K_r \; \vert \; 0 \leq r \leq h)$ be a generic splitting tower of a form $\varphi$ over $k$ and let $(\varphi_r \; \vert \; 0 \leq r \leq h)$
be the associated higher kernel forms of $\varphi$. We fix a number $r$ with $0 < r < h$. We assume that  $\varphi_r$ is definable over $k$. {\it Is there, up to isometry, only one form $\eta$ over $k$ such that  $\varphi_r \cong \eta \otimes K_r$?} $\{$Note that this is trivially true for $r=0$ and $r=h$.$\}$

It is known that the answer is ``yes'' for $\Char k\ne 2$. A proof can be found in  [K$_6$, \S 7]. Fortunately this proof remains valid in characteristic $2$ when the form $\varphi$ is {\it strictly regular}. We will reproduce it below. In slightly more generality we can show:

\begtheorem{2.} Let $\varphi$ be strictly regular. Let  $\eta_1$ and $\eta_2$ be  anisotropic forms over $k$ with $\eta_1 \otimes K_r \sim
\eta_2 \otimes K_r \sim \varphi \otimes K_r$ and $\dim \eta_1 < \dim \varphi_{r-1}$, $\dim \eta_2 < \dim \varphi_{r-1}$. Then  $\eta_1
\cong \eta_2$.
\endtheorem

\begproof{\rm[K$_6$, p.1 ff.].} We assume without loss of generality that the form $\eta_1$ has minimal dimension
among all anisotropic forms $\delta$ over $k$ with   $\delta \otimes K_r \sim \varphi \otimes K_r$. 

We first settle the case $r=1$.
Assume that $\eta_1$ and $\eta_2$ are not isometric. Then $\eta_1 \perp (- \eta_2)$ has a kernel form $\zeta \ne 0$ with
$\zeta \otimes k(\varphi) \sim 0$. By the Norm Theorem and the Subform Theorem there exist a form $\psi$ over $k$ and an
element $a \in k^*$ such that
$$\varphi  \perp  \psi \cong a \zeta.$$
It follows that $\varphi \otimes k(\varphi) \sim (- \psi) \otimes k(\varphi)$. Now 
$$\dim \eta_1 + \dim \eta_2 \geq  \dim \zeta  =  \dim \varphi + \dim \psi,$$
and so
$$\dim \eta_1 - \dim \psi  \geq  \dim \varphi - \dim \eta_2  >  0.$$
This contradicts the minimality of $\dim \eta_1$. Therefore, $\eta_1 \cong \eta_2$.

Assume next that  $r \geq 2$. We proceed by induction on $r$. Assume that $\eta_1$ and $\eta_2$ are not isometric, and thus  $\eta_1 \not\sim \eta_2$. Let $s$ be the largest integer between $0$ and $r-1$ such that $\eta_1 \otimes K_s \not\sim \eta_2\otimes K_s$. If $s>0$ we apply the induction hypothesis for $r-s$ 
to the kernel forms of  $\varphi \otimes K_s$, $\eta_1 \otimes K_s$, $\eta_2 \otimes K_s$, and obtain a contradiction. Hence $s=0$.
For the kernel form  $\zeta$ of $\eta_1 \perp (- \eta_2)$ we have again that $\zeta \otimes k(\varphi) \sim 0$ and, just as in the case $r=1$ above, we obtain a contradiction with the minimality of $\dim\eta_1$.\qed
\endproof

Let $\varphi$ be a neighbour of a Pfister form $\tau$ and let $\eta$ be the complementary form of $\varphi$. By \S20 we have $\varphi
\otimes k(\varphi) \sim (- \eta) \otimes k(\varphi)$. We further assume that $\dim \eta \geq \dim QL(\eta) + 2$. $\{$In the case $\Char k\ne 2$ this implies  $\dim \eta \geq 2$.$\}$ Now we can form the function field $k(\eta)$. We have $k(\varphi) \sim_k k(\tau)$ and $\tau \otimes k(\eta) \sim 0$. Hence there exists a place from 
$k(\varphi)$ to $k(\eta)$ over $k$, and so we also have $\varphi \otimes k(\eta) \sim (- \eta) \otimes k(\eta)$. Conversely we now ask:
\medskip

\noi{\it Let $\varphi$ and $\eta$ be anisotropic forms over $k$ with $\dim \varphi> \dim \eta \geq 2 + \dim QL(\eta)$. Assume that  $\varphi \otimes
k(\varphi) \sim \eta \otimes k(\varphi)$ and $\varphi \otimes k(\eta) \sim \eta \otimes k(\eta)$. Is then $\varphi$ a Pfister neighbour with complementary form $-\eta$?}
\medskip

If $\Char k\ne 2$, this is true, cf. [K$_6$, Th. 8.9]. The proof of this fact again utilizes the argument described in (B), which breaks down in characteristic~$2$. 
\bigskip

\noi{\bf Addendum} (2009) 
\smallskip

\noi Problem (B) is now solved. Hoffmann and Laghribi  have established Theorem~1 above also in characteristic two, without any restrictions [HL], [EKM, \S26]. 

