\titlea{\S32}{Generic Splitting of Stably Conservative Specialized Quadratic Forms}
\leftrunningtitle{Chapter IV. Specialization with Respect to Quadratic Places}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S32 Generic Splitting of Stably Conservative Specialized Quadratic Forms}

For a quadratic form $\varphi$ over a field
$k$, which has   SCR  with respect to a quadratic place $\Gamma
\colon k \to L \cup \infty$, we will show how  the splitting of $\Gamma_*
(\varphi)$ with respect to  
extensions of  the field $L$, conservative for $\Gamma_* (\varphi)$, is regulated by an arbitrarily given generic splitting tower
 $(K_r \; \vert \; 0 \leq r \leq h)$ and quadratic places coming from the fields
$K_r$  (see Theorem~2 below).

For this purpose we need the concept of composition of two quadratic places.

\begdefinition{1.} Let  $\Lambda = (\lambda, H, \chi) \colon K \to
L \cup \infty$ and $M = (\mu, D, \psi) \colon L \to F \cup \infty$ be two
quadratic places. Let 
$$H_0 \colon  =  \{ x \in H, \; \chi (\alpha) \in D \}.$$
Clearly
$$M \circ \Lambda \colon  =  (\mu \circ \lambda, H_0, \psi \circ (\chi \vert
H_0))$$
is a quadratic place from  $K$ to $F$, called the {\it
compositum} of $\Lambda$ and $M$.\Idx{compositum of Q-places@compositum of $Q$-places}
\enddefinition

The theory of these composita is fraught with danger. We will not look at its details, as we would like to advance as quickly as possible towards the core of a generic splitting theory of  stably conservative specialized forms  $\Lambda_* (\varphi)$. This is possible just with Definition~1 and \S 31, although many things would become clearer conceptually after a detailed study of the formal properties of composita of quadratic places. 

We only note the following trivial fact. 

\begremark{} If $\Lambda' \colon K \to L \cup \infty$, $M'
\colon L \to F \cup \infty$ are quadratic places with $\Lambda'
\supset \Lambda$, $M' \supset M$, then  $M' \circ \Lambda' \supset
M \circ \Lambda$.
\endremark

\begtheorem{1.} Let  $\Gamma \colon k \to L \cup \infty$ be a 
quadratic place and $\varphi$ a quadratic form over $k$ which has
SCR with respect to $\Gamma$. Let $r \in {\Bbb N}_0$ and let $K \supset k$
be a field extension which is  $\varphi$-conservative and generic
for the splitting off of $r$ hyperbolic planes. Then the following statements 
are equivalent:
\smallskip	
\setitemindent{$(2)$}

\item{$(1)$} $\ind (\Gamma_* (\varphi)) \geq r$.
\smallskip

\item{$(2)$} There exists a quadratic place $\Lambda \colon K \to L
\cup \infty$ with $\Lambda_* \vert k \supset \Gamma$.
\endtheorem

\begproof{}  (2) $\Rightarrow$ (1): By \S 31, Theorem~6,  $\varphi
\otimes K$ has SCR with respect to $\Lambda$ and $\Lambda_* (\varphi \otimes K)
\cong \Gamma_* (\varphi)$. Now $r \times H < \varphi \otimes K$
and thus, by \S 31, Theorem~5, also $r \times H < \Lambda_* (\varphi
\otimes K)$. Hence $\Gamma_* (\varphi)$ has  Witt index $\geq
r$.

(1) $\Rightarrow$ (2): We carry out the proof in two steps.

(a) We choose a purely transcendental extension $E \supset
k$ and a place $\mu \colon E \to L \cup \infty$ with $\hat{\mu}
\vert k \supset \Lambda$. This is possible by \S30, Theorem~1. Then
we choose a generic splitting field $F
\supset E$ of $\varphi \otimes E$ for the splitting off of $r$
hyperbolic planes. Finally we choose an $s \in
{\Bbb N}_0$ such that $\varphi \perp s \times H$ has CR with respect to
 $\Gamma$. By \S 31, Lemma~1, $(\varphi \perp s \times H)_E =
\varphi_E \perp s \times H$ has FR with respect to $\mu$, and
$$\mu_* (\varphi_E \perp s \times H)  =  \Gamma_* (\varphi \perp s \times H) 
=  \Gamma_* (\varphi) \perp s \times H.$$
$E$ is a generic splitting field of $\varphi_E \perp s
\times H$ for the splitting off of $r + s$ hyperbolic planes.
The form $\Gamma_* (\varphi) \perp s \times H$ over $L$
is nondegenerate since by assumption~(1) its quasilinear part is
anisotropic. Moreover, this form has Witt index $\geq r+s$ by assumption~(1). Thus,
by the generic splitting theory of \S 12, there  exists a place $\sigma
\colon F \to L \cup \infty$ which extends $\mu$. 
For the associated quadratic place  $\hat{\sigma} \colon F \to L \cup
\infty$ we have $\hat{\sigma} \vert E \supset \hat{\mu}$ and thus
$\hat{\sigma} \vert k \supset \hat{\mu} \vert k \supset \Gamma$.  
One should observe now that $F$ is also a generic splitting field
for the splitting off of  $r$ hyperbolic planes from
$\varphi$ (instead of $\varphi_E$).  So for 
 {\it this} field $F$ instead of the given field $K$ we have a proof of statement~(2). 

(b) The fields  $K$ and $F$ are specialization equivalent over $k$.
We choose a place  $\rho \colon
K \to F \cup \infty$ over $k$ and form the compositum $\Lambda =
\hat{\sigma} \circ \hat{\rho} \colon K \to L \cup \infty$. We verify that
$\Lambda \vert k \supset \Gamma$ and are done.

We have  $\hat{\rho} = (\rho, Q ({\nam o}_{\rho}), \check{\rho})$,
$\hat{\sigma} = (\sigma, Q ({\nam o}_{\sigma}), \check{\sigma})$,
where $\check{\rho}$, $\check{\sigma}$ are the characters 
 $Q({\nam o}_{\rho}) \to Q(L)$, $Q ({\nam o}_{\sigma}) \to Q(L)$, induced by
$\rho$, $\sigma$, i.e., $\check{\rho} (\langle a
\rangle) = \langle \rho (a) \rangle$ for $a \in {\nam
o}_{\rho}^*$,  and $\check{\sigma} (\langle b \rangle) = \langle \sigma
(b) \rangle$ for $b \in {\nam o}_{\sigma}^*$. Hence $\Lambda =
(\sigma \circ \rho, H_0, \chi)$ with
$$H_0  =  \{ \langle a \rangle \in Q(K) \; \vert \; a \in {\nam o}_{\rho}^*, \;
\langle \rho (a) \rangle \in Q ({\nam o}_{\sigma}) \}$$
and $\chi = \check{\sigma} \circ (\check{\rho} \vert H_0)$.

Let  $\Gamma = (\gamma, D, \eta)$. For $a \in k^*$ with $\langle a
\rangle \in D$ we have $\langle a \rangle_K \in Q ({\nam o}_{\rho})$ and
$\check{\rho} (\langle a \rangle_K) = \langle a \rangle_F$, since
$\rho$ is a place over $k$.  Since $\hat{\sigma} \vert k
\supset \Lambda$ we have $\langle a \rangle_F \in Q({\nam o}_{\sigma})$,
hence $\langle a \rangle_K \in H_0$ and furthermore
$$\chi (\langle a \rangle_K)  =  \check{\sigma} (\langle a \rangle_F)  =  \eta
(\langle a \rangle).$$
Thus we have verified that  $\Lambda \vert k \supset \Gamma$.\qed
\endproof

As a result of  \S 31 (Theorems 4 and 6) the following corollary of Theorem~1 is
obvious.

\begscholium{} Assume that statements (1) and (2) of the theorem hold. Let\break 
$\varphi \otimes K \cong \psi \perp r
\times H$. Then $\psi$ has SCR with respect to $\Lambda$ and
$$\Gamma_* (\varphi)  \cong  \Lambda_* (\psi) \perp r \times H.\eqno\square$$
\endscholium

Now nothing stands in the way of a generic splitting theory for  $\Gamma_*
(\varphi)$. We note an easy lemma.

\beglemma{1.} Let  $\Lambda \colon K \to L \cup \infty$ be a quadratic place
and $\varphi$ a quadratic form over $K$ which has CR $($resp. SCR$)$ with respect to
$\Lambda$. Further, let  $j \colon L
\hookrightarrow L'$ be a field extension, i.e., a trivial place, which is 
$\Lambda_* (\varphi)$-conservative. Then 
 $\varphi$  also has CR $($resp. SCR$)$ with respect to 
$\hat{j} \circ \Lambda$ and
$$(\hat{j} \circ \Lambda)_* (\varphi)  =  \Lambda_* (\varphi) \otimes_L L'.$$
\endlemma

\begproof{} Let $\Lambda = (\lambda, H, \chi)$. We have $\hat{j} =
(j, Q(L), Q(j))$, where $Q(j)$ denotes the homomorphism from 
$Q(L)$ to $Q(L')$, induced by $j$. We obtain
$$\hat{j} \circ \Lambda  =  (j \circ \lambda, H, Q(j) \circ \chi),$$
and of course  ${\nam o}_{j \circ \lambda} = {\nam
o}_{\lambda}$. This yields the assertion for conservative reduction. 
The extension to the case of SCR is obvious.\qed
\endproof

\begtheorem{2.} Let $\varphi$ be a quadratic form over $k$ and
$\Gamma \colon k \to L \cup \infty$ a quadratic place with respect to which
$\varphi$ has SCR. Let $(K_r \; \vert \; 0 \leq r \leq h)$ be a generic splitting tower
of $\varphi$ with higher indices
$i_r$ and higher kernel forms $\varphi_r$. Finally, let  $L'
\supset L$ be a   $\Gamma_* (\varphi)$-conservative field extension
of $L$. We choose a quadratic place
$\Lambda \colon K_m \to L' \cup \infty$ which extends $\hat{j} \circ \Gamma$
 $($i.e., $\Lambda \vert k \supset \hat{j} \circ \Gamma$$)$, and which in the case
$m < h$ cannot be extended to a quadratic place from 
 $K_{m+1}$ to $L'$. Then
$$\ind (\Gamma_* (\varphi) \otimes L')  =  i_0 + \dots + i_m.$$
The form $\varphi_m$ has SCR with respect to $\Lambda$ and
$$\ker (\Gamma_* (\varphi) \otimes L')  =  \Lambda_* (\varphi_m).$$
\endtheorem

\begproof{} By Lemma~1 we may replace  $\Gamma$ by the quadratic place
 $\hat{j} \circ \Gamma \colon K \to L' \cup
\infty$ and may thus assume without loss of generality that
$L' = L$. We finish by traversing the proof of  \S 12, Theorem 4 once more
and replacing  $\gamma$ by $\Gamma$, $\lambda$ by
$\Lambda$ and FR by SCR everywhere. \qed
\endproof

If $\Char L\ne 2$ then Theorem~2 is essentially the same theorem as 
 \S 30, Theorem~4. We have thus found a new proof for that theorem, which is clearly different from the proof in  \S30. There, a theorem about bad reduction (\S30, Theorem~2) was an important tool, for which we have not developed a counterpart which is valid when $\Char L = 2$. 

We have reached a point where we can envisage applications in the theory of quadratic forms, and in particular in the generic splitting theory, which necessitate the employment of quadratic places, similar to what we did in Chapter~III for ordinary places. This would  go beyond the scope of this book however. From a scientific point of view we would also be threading on virgin soil. To my knowledge there exist only two publications, [K$_7$] and [KR], in which quadratic places are used. Both works assume characteristic $\ne 2$.


