\titlea{\S22}{Pfister Neighbours and Excellent Forms}
\leftrunningtitle{Chapter III. Some Applications}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S22 Pfister Neighbours and Excellent Forms}

In the following  $k$ denotes a field of arbitrary characteristic
and a ``form'' will always mean a {\it \nd\ \q} form.

\begdefinition{1.}{\rm (cf. [K$_6$, Def.7.4] for $\Char k \ne
2$).} A form $\vf$ over $k$ is called a {\it \Pn}\Idx{Pfister neighbour} 
if there exist a
\q\ \Pf\ $\tau$ over $k$ and an element $a \in k^*$ such that
$\varphi < a \tau$ and $\dim \varphi > \frac{1}{2} \dim \tau$.
More precisely we then call $\vf$ a {\it neighbour of the \Pf}
$\tau$.
\enddefinition

In \S16 we already introduced the notion of {\it close \Pn} for
the case $\Char k=2$. Close \Pn s are of course \Pn s in the
current sense.

If $\vf$ is a neighbour of a \Pf\ $\tau$, then we must clearly
have $\dim \varphi \geq 2$, and in case $\dim \varphi = 2$ we must
moreover have $\varphi \cong a \tau$.

\begtheorem{1.} If $\vf$ is a neighbour of a \Pf\ $\tau$ and $a$
is an element of $k^*$ with $\varphi < a \tau$, then the form
$\tau$ and also the form $a \tau$ are up to isometry uniquely
determined by $\vf$. The fields  $k(\varphi)$ and $k (\tau)$ are
specialization equivalent over~$k$.
\endtheorem

\begproof{} The form $a \tau_K$ is hyperbolic over the field $K
\colon = k(\tau)$. Let $E = (E, q)$ be a \qs\ corresponding to $a
\tau_K$ and $F = (F, q \vert F)$ a quadratic subspace of $E$
corresponding to the subform $\varphi_K$ of $a \tau_K$. Let $\dim
E = 2^d$. There exists a vector subspace $U$ of $E$ of dimension
$2^{d-1}$ with $q(U) = 0$. From the obvious inequality
$$\dim (F\cap U) + \dim E \geq \dim F + \dim U$$
and the assumption that $\dim F > 2^{d-1}$ one sees that
$\dim (F \cap U) > 0$, and thus that $F \cap U \ne \{ 0 \}$. Hence $F$
is isotropic and thus the form $\varphi_K = \varphi \otimes
k(\tau)$ is isotropic. Therefore there exists a place from $k(\varphi)$
to $k(\tau)$ over $k$. Since $\vf$ is a subform of $a \tau$, the
form $\tau \otimes k(\varphi)$ is also isotropic. Thus there also
exists a place from $k(\tau)$ to  $k(\varphi)$. (A conclusion we
already reached many times!) Hence $k(\tau) \sim_k k (\varphi)$.

By the corollary of \S21, Theorem~2 it now follows that $\tau$ is
uniquely determined by $\vf$ up to isometry. If $\varphi < a \tau$
and $c \in D(\varphi)$ is arbitrarily chosen, then $a c \in
D(\tau)$, thus $\tau \cong a c \tau$, and so $a \tau \cong c
\tau$. This shows that also the form $a \tau$ is uniquely
determined by $\varphi$ up to isometry.\qed
\endproof

\begdefinition{2.} Let $\vf$ be a \Pn, thus $\varphi < a \tau$
where $\tau$ is a \Pf\ and $\dim \varphi > \frac{1}{2} \dim \tau$.
Then we call the by $\vf$ up to isometry uniquely determined polar
$\Pol_{a \tau} (\varphi)$ of $\varphi$ in $a \tau$ (cf. \S 16) the
{\it complementary form}\Idx{complementary form} 
of $\vf$. Furthermore we call $\tau$ the
{\it neighbouring \Pf} of $\vf$.\Idx{neighbouring Pfister form}
\enddefinition

\begremarks{} (1) Again let $2^d = \dim \tau$. Clearly the
complementary form $\eta = \Pol_{a \tau} (\varphi)$ of $\varphi$
has dimension $2^d - \dim \varphi < 2^{d-1}$. Furthermore
$\varphi$ and $\eta$ have the same quasi-linear part, $QL(\varphi)
= QL(\eta)$, as already observed in \S16, Lemma~1.

(2) In particular $\eta$ is strictly regular if and only if $\vf$
is strictly regular. We then have $\varphi \perp \eta \cong a
\tau$. This is always the case when $k$ has characteristic $\ne
2$. In characteristic~$2$ we also have that $\eta$ is regular if
and only if $\vf$ is regular.

(3) By \S16, $\vf$ is a close \Pn\ if and only if $\eta$ is
quasi-linear. We then have $\eta = QL(\varphi)$. Moreover, if
$\vf$ is \an, then $\vf$ has height~$1$ by \S16. If $\vf$ is
isotropic, then $\vf$ splits.
\endremarks

\begtheorem{2.} Let $\vf$ be an \an\ \Pn\ over $k$ with
complementary form $\eta$. Then $\varphi \otimes k (\varphi) \sim
(- \eta) \otimes k(\varphi)$.
\endtheorem

\begproof{} Let $\tau$ be the neighbouring \Pf\ of $\vf$ and
$\varphi < a \tau$. If $\varphi$ is strictly regular, then $a \tau
\cong \varphi \perp \eta$ and $\tau \otimes k (\varphi) \sim 0$,
and thus certainly $\varphi \otimes k(\varphi) \sim (- \eta)
\otimes k(\varphi)$. This holds in particular if $k$ is of
characteristic $\ne 2$.

Assume now that $\Char k=2$ and that $\vf$ is not strictly
regular.\fonote{The following calculation generalizes the proof of
Theorem~5, $(i)$ -- $(iii)$ in \S16.} Let $\chi$ be the \ql\ part
of $\varphi$, $\chi = QL(\varphi)$. By the scholium in \S16 we
have an orthogonal decomposition $\tau = \mu \; \perp \; \gamma \;
\perp \; \rho$ with $$\chi < \mu, \quad \dim \mu = 2 \dim \chi,
\quad \varphi \; \cong \; \chi \perp \gamma, \quad \eta \; = \;
\chi \perp \rho.$$ With $K \colon = k (\varphi)$ we obtain $$\mu_K
\perp\gamma_K \perp \rho_K \; \sim \; 0,$$ thus $\gamma_K \sim
\mu_K \perp \rho_K$ (note that $\Char k = 2$!), and so $$\varphi_K
\; \sim \; \chi_K \perp \mu_K \perp \rho_K.$$ Let $\chi \cong
[a_1, \dots, a_r]$. Then $$\mu_K \; \cong \; \left[ {a_1 \atop 1}
\; {1 \atop b_1} \right] \perp \cdots \perp \left[ {a_r \atop 1}
\; {1 \atop b_r} \right]$$ with elements $b_i$ of $k$. Hence
$$\chi \; \perp \; \mu \; \cong \; \chi \; \perp \left[ {0 \atop
1} \; {1 \atop b_1} \right] \perp \cdots \perp \left[ {0 \atop 1}
\; {1 \atop b_r} \right] \; \cong \; \chi \; \perp \; r \times
H,$$ and we obtain $$\varphi_K \; \sim \; \chi_K \perp \rho_K \; =
\; \eta_K.\eqno\square$$
\endproof

We return to the theory of generic splitting established in \S12.

For a given form $\vf$, even of small dimension, it is
in general very complicated to explicitly determine the splitting
pattern $\SP (\varphi)$ or even the higher kernel forms.
However, we will now specify a larger class of forms for which it
is possible to do this, the ``excellent forms''. We define a class
of forms by induction on the dimension (cf. [K$_6$, \S 7] for
$\Char k \ne 2$).

\begdefinition{3.} A form $\vf$ is called {\it excellent}\Idx{excellent form} 
if $\dim
\varphi \leq 1$, or $\vf$ is \ql, or $\vf$ is a \Pn\ with
excellent complementary form.
\enddefinition

$\{$Note that if the ground field $k$ has characteristic~$2$ and
if $\dim \varphi \leq 1$, then $\vf$ is of course \ql. In
characteristic $\ne 2$ on the other hand, there are no \ql\ forms
$\varphi \ne 0$.$\}$

Thus, by this definition, a form $\vf$ over $k$ is excellent \iff\
there exists a finite sequence of forms
$$\eta_0 =\varphi, \eta_1, \dots, \eta_t$$
over $k$ such that $\eta_t$ is \ql\ or at most one-dimensional,
and every form $\eta_r$ with $0 \leq r < t$ is a \Pn\ with
complementary form $\eta_{r+1}$. We call the forms $\eta_r$ the
{\it higher complementary forms} of $\vf$, more specifically we
call $\eta_r$ the {\it $r$-th complementary form}\Idx{higher complementary form}\Idx{r-th complementary form@$r$-th complementary form} 
of $\vf$
$(0 \leq r \leq t)$.

\begtheorem{3.} Let $\vf$ be an \an\ excellent form over $k$ and
let $(\eta_i \; \vert \; 0 \leq i \leq t)$ be the sequence of
higher complementary forms of $\vf$. For every $r \in \{ 0, 1,
\dots, t \}$, let $K_r$ denote the free composite of the fields
$k(\eta_i)$ with $0 \leq i < r$ over $k$. $\{$Read $K_0 = k$ for
$r = 0$.$\}$

Claim: $(K_r \; \vert \; 0 \leq r \leq t)$ is a generic splitting
tower of $\vf$. For every $r \in \{ 0, 1, \dots, t \}$, $\varphi
\otimes K_r$ has kernel form $(-1)^r \eta_r \otimes K_r$. In
particular, $\vf$ has height $t$.
\endtheorem

\begproof{} By induction on $r$. For $r=0$ we don't have to show
anything, so suppose that $r>0$. Let $\tau_0$ and  $\tau_1$ be the
neighbouring \Pf s of $\varphi = \eta_0$ and  $\eta_1$
respectively. Thus there are elements $a_0, a_1$ in $k^*$ with
$\varphi < a_0 \tau_0, \; \eta_1 < a_1 \tau_1$, $\eta_1 =
\Pol_{a_0 \tau_0} (\varphi)$. Furthermore,  $K_1 = k(\varphi)$. By
Theorem~2 we have $\varphi \otimes K_1 \sim (- \eta_1) \otimes
K_1$, and by Theorem~1 we have $k(\varphi) \sim_k k (\tau_0)$, $\;
k(\eta_1) \sim_k k (\tau_1)$.

If $\eta_1 \otimes K_1$ were isotropic, then $\tau_1 \otimes k
(\tau_0)$ would also be isotropic, and thus hyperbolic. This
implies, by \S21, Theorem~1 (or \S21, Theorem~2), that the form
$\tau_1$ is divisible by $\tau_0$. However, this is absurd since
$\tau_1$ has smaller dimension than $\tau_0$. Hence $\eta_1
\otimes K_1$ is \an. Thus $(- \eta_1) \otimes K_1$ is the kernel
form of $\varphi \otimes K_1$.

By definition of excellent forms it is clear that $\eta_1 \otimes
K_1$ is excellent and that for every $r \in \{ 1, \dots, t \}$ the
form $\eta_r \otimes K_1$ is the $(r-1)$-st complementary form of
$\eta_1 \otimes K_1$. If we now apply the induction hypothesis to
$\eta_1 \otimes K_1$ we obtain the complete statement of the
theorem. With regards to this, one should keep in mind that $K_1
(\eta_i \otimes K_1)$ is the free composite of $K_1$ and $k
(\eta_i)$ over $k$  $(1 \leq i \leq t)$ and consequently that
$K_r$ is the free composite of the fields $K_1 (\eta_i \otimes
K_i)$ with $1 \leq i < r$ over $K_1$  $(1 \leq r \leq t)$.\qed
\endproof

In which dimensions do there exist, for a given field $k$,
anisotropic excellent forms over $k$? We want to discuss this
question in the (difficult) case when $k$ has characteristic~$2$.

\begscholium{} Let $\Char k=2$. If $\vf$ is an \an\ excellent form
of dimension $n$ over $k$, then the neighbouring \Pf\ $\tau$ of
$\vf$ is again \an, and $\dim \tau$ is clearly the smallest
$2$-power $2^d \geq n$.

Conversely, let $\tau$ be an \an\ \Pf. In this case we construct
an excellent form with neighbouring form $\tau$ as follows: We
choose a sequence $(\tau_r \; \vert \; 1 \leq r \leq t)$ of
Pfister subforms of $\tau$ with $\tau_1 = \tau$, $\tau_r \; \vert
\; \tau_{r-1}$ for $2 \leq r \leq t$, and a \ql\ subform $\eta$ of
$\tau_t$ with $\dim \eta < \frac{1}{2} \dim \tau_t$, which is
always possible. $\{$$t\geq 1$. It is allowed that $\eta = 0$.$\}$
Then we define successive forms $\eta_t, \eta_{t-1}, \dots,
\eta_0$ in such a way that $$\eta_t = \eta \; , \; \eta_r < \tau_r
\; , \; \eta_{r-1} < \tau_r \; , \; \eta_{r-1} = \Pol_{\tau_r}
(\eta_r) \quad \hbox{for} \quad 1 \leq r \leq t.$$ Now $\varphi
\colon = \eta_0$ is an excellent form with neighbour $\tau$ and
higher complementary forms $\eta_r \; (0 \leq r \leq t)$. The form
$\vf$ is \an\ and has \ql\ part $\eta$. Up to a scalar factor one
can obtain every \an\ excellent form over $k$ in this manner.\qed
\endscholium

Next we show that over suitable fields of characteristic~$2$ there
exist \an\ \Pf s of arbitrarily high dimension.

\begtheorem{4.} Let $\kappa$ be any field and $k = \kappa (u, u_1,
\ldots, u_n)$, where $u, u_1, \ldots, u_n$ is a tuple of unknowns.
The \Pf\ $\langle 1, u_1 \rangle \otimes \cdots \otimes \langle 1,
u_n \rangle \otimes \left[ {1 \atop 1} \; {1 \atop u} \right]$
over $k$ is \an.
\endtheorem

\begproof{} We assume that $\kappa$ has characteristic~$2$ and
leave the proof for the case $\Char \kappa \ne 2$ to the Reader. We work
over the iterated power series field
$$K \colon = \kappa ((u)) ((u_1)) \dots ((u_n)),$$
which contains $k$. Let $\tau$ be the \Pf\ from above, considered
as a form over $K$. We  show by induction on $n$ that $\tau$ is
\an.
\medskip

\noi${n = 0}:$ The equation $u = x^2 - x$ is with $x \in \kappa
((u))$ unsolvable, so that the form $\left[ {1 \atop 1} \; {1
\atop u} \right]$ is \an\ over $\kappa ((u))$.
\medskip

\noi${n-1 \to n:}$ Let $F \colon = \kappa ((u)) ((u_1)) \dots
((u_{n-1}))$. We have $\tau = \rho_K \perp u_n \rho_K$ with $\rho$
an \an\ \Pf\ over $F$ by the induction hypothesis. There is a
place $\lambda \colon K \to F \cup \infty$ over $F$ belonging to
the obviously discrete valuation of $K$ over $F$ with prime
element $u_n$. The form $\rho_K$ has good reduction \wrtl\  and
$\lambda_* (\rho_K) = \rho$. We further have that $\lambda (u_n
c^2) = 0$ or $= \infty$ for every $c \in K^*$. If $\tau$ were
isotropic, and thus hyperbolic, then we would have $\rho_K \cong
u_n \rho_K$. Applying $\lambda_W$ would then give $\rho \sim 0$, a
contradiction. Therefore $\tau$ is anisotropic.\qed
\endproof

\noi {\bf Example.} We choose two natural numbers $n > m$, and a
sequence of natural numbers $$s_1 = n > s_2 > \cdots > s_t = m.$$
Let $k = \kappa (u, u_1, \dots, u_n)$, as in Theorem~4, and
$\Char\kappa = 2$. We form the -- by Theorem~4 \an\ --  \Pf s
$$\tau_r \colon = \langle 1, u_1 \rangle \otimes \cdots \otimes
\langle 1, u_{s_r} \rangle \otimes \left[ {1 \atop 1} \; {1 \atop
u} \right]$$ $(1 \leq r \leq t)$, and then the \ql\ form $$\eta
\colon = [1, u_1, u_2, \dots, u_s]$$ for some $s < s_t = m$. We
have $\eta < \tau_t$ and $\dim \eta < \frac{1}{2} \dim \tau_t$.
Using the procedure outlined in the scholium above we obtain an
\an\ excellent form $\vf$ of height $t$ over $k$ with
$(s+1)$-dimensional \ql\ part~$\eta$.\qed
\medskip

We utilize Theorem~4 and this example to illustrate principles
about the specialization of \qf s, obtained earlier (\S15,
Theorem~5, \S11, Theorem~2). We consider the question of when an
excellent form has good reduction \wrt\ a place, and   its
specialization is again excellent.

Theorem~4 gives us the ``generic'' $(n+1)$-fold \Pf\ of a field
$\kappa$. Namely, if $\rho$ is an $(n+1)$-fold \Pf\ over $\kappa$,
and if we have chosen a representation
$$\rho \cong \langle 1, a_1
\rangle \otimes \cdots \otimes \langle 1, a_n \rangle \otimes
\left[ {1 \atop 1} \; {1 \atop a} \right]$$
$(a \in \kappa$, all $a_i \in \kappa)$, then there exists a place
$\lambda \colon \kappa (u, u_1, \dots, u_n) \to \kappa
\cup \infty$ over $\kappa$ with $\lambda (u) = a$, $\lambda (u_i) =
a_i$ for $i = 1, \dots, n$. The form
$$\tau \colon = \langle 1, u_1 \rangle \otimes \cdots \otimes
\langle 1, u_n \rangle \otimes \left[ {1 \atop 1} \; {1 \atop u}
\right]$$
has good reduction with respect to every such place
$\lambda$, and $\lambda_* (\tau) \cong \rho$.

Now suppose that $\rho$ is \an. We choose a binary quadratic
subform $\gamma$ of $\tau$ with $1 \in D (\gamma)$, which has good
reduction \wrtl, e.g. $\gamma = \left[ {1
\atop 1} \; {1 \atop u} \right]$. If $\sigma$ is a \Pf\ over $k$
with $\gamma < \sigma < \tau$, then $\sigma$ is clearly obedient
\wrtl. By \S15, Theorem~5, and \S11, Theorem~2, it follows that
$\sigma$ has good reduction \wrtl\ and that $\lambda_* (\sigma)$
is a subform of $\rho = \lambda_* (\tau)$. Furthermore,
$\lambda_* (\gamma)$ is a subform of  $\lambda_*(\sigma)$.

We want to obtain that $\lambda_* (\sigma)$ is again a \Pf. By
\S12, $\lambda_* (\sigma)$ has height $\leq 1$. Since $\lambda_*
(\sigma)$ is anisotropic, and thus doesn't split, we must have $h
(\lambda_* (\sigma)) = 1$. Furthermore, $[1] < \sigma$ implies,
again by \S15, Theorem~5, etc. that $[1] = \lambda_* ([1])$ is a
subform of $\lambda_* (\sigma)$, thus $1 \in D (\lambda_*
(\sigma))$. Finally \S20, Theorem~5, tells us that $\lambda_*
(\sigma)$ is a \Pf. It divides $\lambda_* (\tau) = \rho$.

We turn our attention to the construction of excellent forms in
the scholium above. There we choose the \Pf\ $\tau_t$ so that
$\gamma < \tau_t$. Now all forms $\tau_r$ have good reduction
\wrtl, every $\lambda_* (\tau_r)$ is a \Pf, and $\lambda_*
(\tau_r) \; \vert \; \lambda_* (\tau_{r-1})$ for $2 \leq r \leq
t$. The \ql\ form $\eta$ is automatically obedient \wrtl. Thus, by
\S15, Theorem~5, etc., it has good reduction \wrtl\ and $\lambda_*
(\eta)$ is a subform of $\lambda_* (\tau_t)$.

If the complementary form $\eta_{t-1}$ of $\eta = \eta_t$ in
$\tau_t$ is obedient \wrtl, then we obtain in the same fashion
that $\eta_{t-1}$ has good reduction \wrtl, and that $\lambda_*
(\eta_{t-1}) < \lambda_* (\tau_t)$. Furthermore, $\lambda_*
(\eta_{t-1})$ is now orthogonal to $\lambda_* (\eta_t)$, and by
dimension considerations it is now clear that $\lambda_*
(\eta_{t-1})$ is the polar of $\lambda_* (\eta_t)$ in $\lambda_*
(\tau_t)$.

We can continue this line of reasoning in so far that it is
guaranteed that all $\eta_r$ are obedient \wrtl, and obtain
finally that the excellent form $\varphi = \eta_0$ has good
reduction \wrtl, and that $\lambda_* (\varphi)$ is an excellent
subform of $\lambda_* (\tau) = \rho$ with $\rho$ as neighbour.

If one considers $\gamma = \left[ {1 \atop 1} \; {1 \atop u}
\right]$, and starts with $\eta = [1, u_1, u_2, \dots, u_s]$, as
in the example above,  one can reason inductively that all
$\eta_r$ are  indeed obedient \wrtl, and thus actually have good
reduction \wrtl.

Now, if $\rho$ is anisotropic, then it follows from \S15,
Theorem~5, that every subform $\sigma$ of $\tau$ has good
reduction \wrtl, and that $\lambda_* (\sigma)$ is a subform of
$\rho$. If $\sigma$ is a \Pf, then $\lambda_* (\sigma)$ has height
$\leq 1$ by \S12. Since $\lambda_* (\sigma)$ is anisotropic, and
so doesn't split, $\lambda_* (\sigma)$ must have height exactly
$1$. Moreover, $\lambda_* (\sigma)$ is strictly regular, since
this is the case for $\sigma$.

Furthermore, $[1] < \sigma$ implies that $[1] = \lambda_* [1]$ is
a subform of $\lambda_* (\sigma)$. Now we conclude with \S20,
Theorem~5, that $\lambda_* (\sigma)$ is a \Pf\ (and it divides the
form $\rho$ since it is a subform of $\rho$).

Every Pfister subform $\sigma$ of $\tau$ thus has good reduction
\wrtl, and provides a Pfister subform $\lambda_* (\sigma)$ of
$\rho$. By the scholium above it is now clear that the \an\
excellent form $\vf$ in our example above
has good reduction \wrtl, and that $\lambda_* (\varphi)$ is a
neighbour of the \Pf~$\rho$.

Nevertheless we cannot consider $\vf$ to be a ``generic''
excellent form of height $1$ with particular splitting pattern for
$\kappa$ since there are too many other possibilities to construct
excellent subforms of $\tau$. Our reasoning about the
specialization of excellent forms remains valid in the general
situation however. We record

\begtheorem{5.} Let $\lambda \colon K \to L \cup \infty$ be a
place and $\tau$ a \Pf\ which has good reduction \wrtl. Assume
that the form $\lambda_* (\tau)$ is anisotropic. Then $\lambda_*
(\tau)$ is again a \Pf. Every excellent subform $\vf$ of $\tau$
which is a neighbour of $\tau$ has good reduction \wrtl, and
$\lambda_* (\varphi)$ is an excellent form, neighbouring
$\lambda_* (\tau)$, of the same height as $\vf$.\qed
\endtheorem

In the situation of the last theorem, the form $\lambda_* (\tau)$
is furthermore also a \Pf\ when it is isotropic, since it follows
already from the generic splitting theory in \S9 that $\lambda_*
(\tau) \cong 2^{d-1} \times H$ with $2^d = \dim \tau$.
