\titlea{\S14}{Generic Splitting Towers of a Specialized Form}
\leftrunningtitle{Chapter II. Generic Splitting Theory}
%\leftrunningtitle{Specialization of Quadratic and Symmetric
%Bilinear Forms}
\rightrunningtitle{\S14 Generic Splitting Towers
of a Specialized Form}

In the following, let $\gamma \colon k \to L \cup \infty$ be a
place and $\vf$ a \qf\ over $k$ having FR \wrt\ $\gamma$. Assume
that the form
$\overline{\varphi} \colon = \gamma_* (\varphi)$ over $L$
is \nd. Finally, let $(K_r \; \vert \; 0 \leq r \leq h)$ be a
generic splitting tower of $\vf$.

We have seen in \S12 that the splitting behaviour of
$\overline{\varphi}$ \wrt\ $\overline{\varphi}$-conservative
extensions of the field $L$ is controlled by the tower
$(K_r \; \vert \; 0 \leq r \leq h)$, cf. \S12, Theorem~4 and
Scholium~2. Hence we may hope that it is actually possible to
construct a generic splitting tower of $\overline{\varphi}$
in a natural way from $(K_r \; \vert \; 0 \leq r \leq h)$. We will
devote ourselves to the task of constructing such a tower.

In the special case where the place $\gamma$ is trivial and the
tower $(K_r\; \vert\; 0 \leq r \leq h)$ is regular, we already dealt
with this task in the previous section (\S13, Theorem~3). In
general we cannot expect to find a solution as nice as the
solution in \S13.

Let us continue by fixing some notation. Let $i_r$ be the higher
indices and $\vf_r$ the higher kernel forms of $\vf$ \wrt\ the
given splitting tower $(K_r \; \vert \; 0 \leq r \leq h)$ of
$\varphi$. Thus we have
$\SP (\varphi) = \{ i_0 + \dots + i_r \; \vert \; 0 \leq r \leq h
\}$. Further, let $J$ be the subset of $\{ 0, 1, \dots, h \}$ with
$\SP (\overline{\varphi}) = \{ i_0 + \dots + i_r \; \vert \; r \in J
\}$. The set $J$ contains $e \colon = h (\overline{\varphi})$
elements. Let us list them as
$$0\leq t(0) < t(1) < \dots < t(e) = h.$$
We formally set $t(-1) = -1$.

Let $(L_s \; \vert \; 0 \leq s \leq e)$ be a generic splitting
tower of $\overline{\varphi}$. Following \S12, Scholium~2 we
choose places
$\mu_s \colon K_{t(s)} \to L_s \cup \infty \quad (0 \leq s \leq e)$
such that $\mu_0$ extends the place $\gamma$ and such that every
$\mu_s$ with $1 \leq s\leq e$ extends the place $\mu_{s-1}$. For
every $r \in \{ 0, \dots, h \}$ we have an
$s \in \{ 0, \dots, e \}$ with $t(s-1) < r \leq t(s)$. Let
$\lambda_r \colon K_r \to L_s \cup \infty$ be the restriction of
the place $\mu_s$ to $K_r$. $\{$In particular, $\mu_s = \lambda_{t(s)}$.$\}$
Let $\psi_s =\ker (\overline{\varphi} \otimes L_s)$ be the $s$-th
higher kernel form of $\overline{\varphi}$. By \S12
$\varphi_{t(s)}$ has FR \wrt\ $\mu_s$ and $(\mu_s)_* (\varphi_{t(s)}) \cong
\psi_s$. Furthermore, for $t(s-1) < r < t(s)$,
$\varphi_r$ also has FR \wrt\ $\lambda_r$ and
$$(\lambda_r)_* (\varphi_r) \cong \psi_s \, \perp \, (i_{r+1} + \dots +
i_{t(s)}) \times H.$$

Given a place $\alpha \colon E \to F \cup \infty$ and a subfield
$K$ of $E$, we denote the image of $K$ under $\alpha$ in general
sloppily with $\alpha(K)$, in other words,
$$\alpha (K) \colon = \alpha ({\nam o}_{\alpha} \cap K).$$

In the important case that the place $\gamma$ is surjective, i.e. $\gamma
(k) = L$, we would like to choose the places $\mu_s$ in such a way
that $(\mu_s (K_{t(s)}) \; \vert \; 0 \leq s \leq e)$ is a generic
splitting tower of $\overline{\varphi}$. For not necessarily
surjective $\gamma$, one could try to achieve that
$(L \cdot \mu_s (K_{t(s)}) \; \vert \; 0 \leq s \leq e)$ is a
generic splitting tower of $\overline{\varphi}$, where
$L \cdot \mu_s (K_{t(s)})$ denotes the subfield of $L_s$ generated
by $L$ and $\mu_s (K_{t(s)})$ in $L_s$.\fonote{Now we are {\it
not} using the notation established in \S13, Definition~6.} In any
case, for arbitrary choice of $\mu_s$ -- as above -- the following
theorem holds.

\begtheorem{1.} For $t(s-1) < r \leq t(s)$ the subfield $\mu_s (K_r) \cdot L$ of
$L_s$, generated by $\mu_s (K_r)$ and $L$, is a generic splitting
field of $\overline{\varphi}$ of level $s$.
\endtheorem

\begproof{} Let $F_r \colon = \lambda_r (K_r) \cdot L$. By
restricting the range, we obtain from $\lambda_r$ a place from $K_r$ to
$F_r$ which extends $\gamma$. Hence
$$\ind (\overline{\varphi} \otimes F_r) \geq \ind (\varphi \otimes K_r) = i_0
+ \dots + i_r.$$
Since $\ind (\overline{\varphi} \otimes F_r)$ is a number of $\SP
(\overline{\varphi})$, we get
$$\ind (\overline{\varphi} \otimes F_r) \geq i_0 + \dots + i_{t(s)}.$$
Thus there is a place from $L_s$ to $F_r$ over $L$. Since $F_r \subset
L_s$, we get $F_r \sim_L L_s$. Thus $F_r$ is a generic splitting
field of $\overline{\varphi}$ of level $s$.\qed
\endproof

This theorem does {\it not} imply that $(K_{t(s)} \cdot L \; \vert \; 0 \leq
s \leq e)$ is a generic splitting tower of $\overline{\varphi}$.
However, in the following we will produce more special situations
for which this {\it is} the case.

\begtheorem{2.} There exists a regular\fonote{cf. \S13,
Definition~5.}
generic splitting tower $(L_s
\; \vert \; 0 \leq s \leq e)$ of $\overline{\varphi}$ and a place
$\lambda \colon K_h \to L_e \cup \infty$ having the following
properties:
\medskip

\setitemindent{(iii)}
\item{$(i)$} $\lambda$ extends $\gamma$.
\medskip
\item{$(ii)$} For every $s \in \{ 0, 1, \dots, e \}$ and every $r$ with $t(s-1) < r \leq
t(s)$ we have $L_s = \lambda (K_r) \cdot L$.
\medskip
\item{$(iii)$} $L_0=L$.
\endtheorem

\begproof{} We inductively construct fields $L_s \supset L$ and
places $\mu_s \colon K_{t(s)} \to L_s \cup \infty$ as follows.
\medskip

\noindent $s = 0$: We let $L_0:=L$. We choose for $\mu_0$ any place from
$K_{t(0)}$ to $L$ which extends $\gamma$. This is possible by
\S12, Scholium~2. Obviously we have $\mu_0 (K_r) \cdot L = L$ for every $r$
with $0 \leq r \leq t(0)$.
\medskip

\noindent $s \to s+1$: By the induction hypothesis, $(L_j \; \vert \; 0 \leq j
\leq s)$ is the beginning of a generic splitting tower of $\overline{\varphi}$.
In particular, $L_{s}$ is a generic splitting field of
$\overline{\varphi}$ of level $s$. Furthermore there is a place
$\mu_{s} \colon K_{t(s)} \to L_{s} \cup \infty$. According to \S12
the form $\varphi_{t(s)}$ has FR \wrt\ $\mu_s$ and $(\mu_{s})_*
(\varphi_{t(s)}) \cong \psi_{s}$.

Now we extend $\mu_{s}$ to a place $\tilde{\mu}_{s}$ from
$K_{t(s)} (\varphi_{t(s)})$ to $L_{s} (\psi_{s})$ (cf. \S9, the
proof of Theorem~3). Let $u \colon = t(s) + 1$ and let $\alpha \colon K_u \to
K_{t(s)} (\varphi_{t(s)}) \cup \infty$ be a place over $K_{t(s)}$,
which does indeed exist since $K_u$ is a generic zero field of
$\varphi_{t(s)}$. Then we have a place $\tilde{\mu}_{s} \circ \alpha$
from $K_u$ to $L_{s} (\psi_{s})$ and define $L_{s+1}$ to be the
composite of the fields $L_{s}$ and $(\tilde{\mu}_{s} \circ \alpha) (K_u)$ in $L_{s}
(\psi_{s})$,
$$L_{s+1} \colon = L \cdot (\tilde{\mu}_{s} \alpha) (K_u).$$

Let $\nu_s \colon K_u \to L_{s+1} \cup \infty$ be the place
obtained from $\tilde{\mu}_{s} \circ \alpha$ by restricting the
range. We have
$$\ind (\overline{\varphi} \otimes L_{s+1}) \geq \ind (\varphi \otimes K_u) = i_0
+ \dots + i_u > i_0 + \dots + i_{t(s)},$$
and thus $\ind (\overline{\varphi} \otimes L_{s+1}) \geq i_0 + \dots +
i_{t(s+1)}$. This implies
$$\ind (\psi_{s} \otimes L_{s+1}) \geq i_u + \dots + i_{t(s+1)} >
0.$$
Thus $\psi_{s} \otimes L_{s+1}$ is isotropic. Since $L_{s+1} \subset L_{s}
(\psi_s)$, we conclude that $L_{s+1}$ is a generic zero field of
$\psi_s$. By \S12, Scholium~2(3) we now have that $\nu_s$ can be
extended to a place from $K_{t(s+1)}$ to $L_{s+1}$. We choose $\mu_{s+1}
\colon K_{t(s+1)} \to L_{s+1} \cup \infty$ to be such a place.
Since $L_{s+1} =
\nu_{s} (K_u) \cdot L$, we have  $L_{s+1} = \mu_{s+1} (K_r) \cdot L$ for every $r$ with $u
\leq r \leq t(s)$.

Eventually our construction yields a generic splitting tower
$$(L_s
\; \vert \; 0 \leq s \leq e)$$
of $\overline{\varphi}$ and a place
$\lambda \colon = \mu_e$ from $K_h$ to $L_e$ which satisfy
properties $(i)$ and $(ii)$ of the theorem.

Let us check that the tower $(L_s \; \vert \; 0 \leq s \leq e)$ is
regular! For $0 \leq s \leq e-1$ our construction gives
$L_{s} \subset L_{s+1} \subset L_{s} (\psi_{s})$. If $\dim \psi_{s} >
2$, then $L_{s} (\psi_{s})$ is regular over $L_{s}$, and thus
$L_{s+1}$ is regular over $L_{s}$. If $\dim \psi_{s} =
2$, then we must have $s = e-1$ and $t(s) = h-1$. Next, we modify
the last step of the construction above in the sense that we
replace $K_{h-1} (\varphi_{h-1})$ by the algebraic closure $E$ of
$K_{h-1}$ in this field and also $K_{e-1} (\psi_{e-1})$ by the
algebraic closure $F$ of $K_{e-1}$ in $K_{e-1} (\psi_{e-1})$.

Let $\mu \colon = \mu_{h-1}$. We have $\varphi_{h-1} \cong \langle a \rangle
\otimes \left[ {1 \atop 1} \; {1 \atop b} \right]$ with $a \in {\nam
o}_{\mu}^*$, $b \in {\nam o}_{\mu}^*$, and $\psi_{e-1} \cong \langle
\overline{a} \rangle \otimes \left[ {1 \atop 1} \; {1 \atop \overline{b}}
\right]$ with $\overline{a} \colon = \mu (a)$, $\overline{b} \colon = \mu
(b)$. We obtain $E = K_{h-1} (\xi)$ and $F = L_{e-1} (\eta)$ with
generators $\xi$, $\eta$ which satisfy the minimal equations $\xi^2 + \xi + b = 0$, $\eta^2 +
\eta + \overline{b} = 0$. The extension  $K_h/E$ is inessential.
This can be verified  as in the proof of \S13, Theorem~3 towards
the end. (There for the case $\dim \varphi = 2$.)

The place $\mu = \mu_{e-1}$ from $K_{h-1}$ to $L_{e-1}$ has
exactly one extension $\tilde{\mu}_{e-1} \colon E \to F \cup \infty$
with $\tilde{\mu}_{e-1} (\xi) = \eta$ (cf. \S 9, Lemma~2). If we
prepend a place from $K_h$ to $E$ over $E$ to $\tilde{\mu}_{e-1}$,
we obtain a place $\mu_e \colon K_h \to F \cup \infty$ which
extends $\mu_{e-1}$. The field $L_e
\colon = \mu_e (K_h) \cdot L$ differs from $L_{e-1}$ since it
splits the form $\overline{\varphi}$ totally. Hence we must have
$L_e = F$. Our generic splitting tower $(L_s \; \vert \; 0 \leq s \leq
e)$ is regular.\qed
\endproof

What we have obtained by now can perhaps best be understood in the
important special case where the place $\gamma \colon k \to L \cup
\infty$ is surjective (i.e. $\gamma (k) = L$). For the given
generic splitting tower $(K_r \; \vert \; 0 \leq r \leq h)$ of
$\varphi$ we have constructed a place $\lambda \colon K_h \to L_e \cup
\infty$ which extends $\gamma$ and for which the tower $(\lambda (K_r) \; \vert \; 0 \leq r
\leq h)$ is  an almost regular generic splitting tower of
$\overline{\varphi}$ ``with repetitions'', i.e. a generic splitting
tower in which the storeys  are listed possibly more than once:
$\lambda (K_r) = \lambda (K_{t(s)})$ for $t(s-1) < r \leq t(s)$.
The place $\lambda$ furnishes a connection between the tower $(K_r
\vert\, 0 \leq r \leq h)$ and the generic splitting tower $(L_s \vert 0 \leq
s \leq e)$ of $\overline{\varphi}$, $L_s = \lambda (K_{t(s)})$
which reflects in an obvious way how the splitting behaviour of
$\overline{\varphi}$ over $L$ is a coarsening   of the splitting
behaviour of $\vf$ over $K$.

In many situations it is more natural or, for some given problem,
more useful (see e.g. [KR]) to  associate to the tower $(K_r \; \vert \; 0 \leq r \leq
h)$ a different generic splitting tower of $\overline{\varphi}$.
Such a situation (with $\gamma$ trivial) was depicted in \S13.
Next we give a further construction which applies to an arbitrary
place $\gamma$ \wrt\ which $\vf$ has FR and with
$\overline{\varphi} = \gamma_* (\varphi)$ \nd, as above.

Specifically, let $(K_r \; \vert \; 0 \leq r \leq h)$ be the
projective standard tower of $\vf$ (see \S13, Definition~4). We
define a field tower $(L_r \; \vert \; 0 \leq r \leq h)$ and a
sequence of places $(\lambda_r \colon K_r \to L_r
\cup \infty \; \vert \; 0 \leq r \leq h)$ with $L_r \supset L$ and $L_r =
\lambda_r (K_r) \cdot L$ for all $r$, $0 \leq r \leq h$, as
follows.\fonote{The equation $L_r = \lambda_r (K_r) \cdot L$ only
signifies that $L_r$ is generated as a field by both subfields
$\lambda_r (K_r)$ and $L$.}

We start with $L_0 \colon = L$, $\lambda_0 \colon = \gamma$. If
$\lambda_r \colon K_r \to L_r \cup \infty$ is already defined and $r <
h$, then $\varphi_r$ has FR \wrt\ $\lambda_r$ by \S12. Let $\overline{\varphi}_r \colon
= (\lambda_r)_* (\varphi_r)$. We set $L_{r+1} \colon = L_r \{
\overline{\varphi}_r \}$. Let $\tilde{\lambda}_r \colon K_r (\varphi_r) \to
L_r (\overline{\varphi}_r) \cup \infty$ be an extension of the place
$\lambda_r$, obtained using the procedure in the proof of \S9,
Theorem~3.

Next, let $\overline{\varphi}_r \not \cong H$. The place
$\tilde{\lambda}_r$ maps $K_{r+1} = K_r \{ \varphi_r \}$ into
$L_{r+1} \cup \infty$. From the construction in \S9 it is clear
that $L_{r+1} = \tilde{\lambda}_{r} (K_{r+1}) \cdot L$. We define
$\lambda_{r+1} \colon K_{r+1} \to L_{r+1} \cup \infty$ by
restricting the place $\tilde{\lambda}_r$. We have $L_{r+1} =
\lambda_{r+1} (K_{r+1}) \cdot L$.

Finally, let $\overline{\varphi}_r \cong H$ (and so $r = h-1$).
Now we have $L_{r+1} = L_r$. From the construction in \S9, we have
$\tilde{\lambda}_r (K_r (\varphi_r)) \cdot L = L_r$. Again we
define $\lambda_{r+1}$, i.e. $\lambda_h$, to be the restriction of
$\tilde{\lambda}_r$ to a place from $K_{r+1}$ to $L_{r+1} = L_r$.
Then, clearly, $\lambda_{r+1} (K_{r+1}) \cdot L = L_{r+1}$.

\begdefinition{} We call the tower $(L_r \; \vert \; 0 \leq r \leq
h)$ the {\it transfer} of the projective standard tower by the place
$\gamma$ and we call $\lambda \colon = \lambda_h \colon K_h \to L_h \cup
\infty$ a {\it transferring place}.
\enddefinition

\begremark{} By the remark in \S9, following the proof of
Theorem~3,
the places $\lambda_r \colon K_r \to L_r \cup
\infty$ for $0 \leq r \leq h-1$ in the construction above are uniquely determined
by $\gamma$. If $\overline{\varphi}_{h-1} \not\cong H$ this is
also the case for $\lambda_h =\lambda$. If $\overline{\varphi}_{h-1} \cong
H$, then there are at most two possibilities for $\lambda_h$,
since $K_h$ is a quadratic extension of $K_{h-1}$ in this
situation. The fields $L_r = \lambda_r (K_r) \cdot L$ are all
uniquely determined by $\varphi$ and $\gamma$.
\endremark

\begtheorem{3.} Let $(L_r \; \vert \; 0 \leq r \leq h)$ be the
transfer of the projective standard tower $(K_r \; \vert \; 0 \leq r \leq
h)$ of $\vf$ by the place $\gamma \colon k \to L \cup \infty$ and
let $\lambda \colon K_h \to L_h \cup \infty$ be a transferring
place. Let $\overline{\varphi}_r
\colon = (\lambda_r)_* (\varphi_r)$ be the specialization of
$\varphi_r$ \wrt\ the restriction $\lambda_r \colon K_r \to L_r \cup \infty$ of $\lambda \quad
(0 \leq r \leq h)$.

Claim: $(L_r \; \vert \; r \in J)$ is a generic splitting tower of
$\overline{\varphi}$ and for every $r \in J$ we have
$\overline{\varphi}_r = \ker (\overline{\varphi} \otimes L_r)$.
For $r\not\in J$, $L_{r+1}/L_r$ is purely transcendental. In
particular, the extension $L_{t(0)}/L$ is purely transcendental.
\endtheorem

\begproof{} Let $r \in \{ 0, \dots, h \}$ be given. From the Witt
decomposition
$$\varphi \otimes K_r \cong (i_0+ \dots + i_r) \times H \, \perp \,
\varphi_r$$
we obtain
$$\overline{\varphi} \otimes L_r \cong (i_0 + \dots + i_r) \times H \; \perp
\; \overline{\varphi}_r,$$
since $(\lambda_r)_* (\varphi \otimes K_r) \cong
\overline{\varphi} \otimes L_r$.

If $r \not\in J$, then $\ind (\overline{\varphi}
\otimes L_r) > i_0 + \dots + i_r$. Now $\overline{\varphi}_r$ is
isotropic and hence $L_{r+1} = L_r \{ \overline{\varphi}_r \}$ is
purely transcendental over $L_r$.

Let a number $s$ be given with $0 \leq s \leq e$. Then $L_{t(s) +1} =
L_{t(s)} \{ (\overline{\varphi}_{t(s)}) \}$ and
$L_{t(s+1)}/L_{t(s)+1}$ is purely transcendental. Hence
$L_{t(s+1)}/L_{t(s)}$ is a generic zero field of
$\overline{\varphi}_{t(s)}$. Furthermore,
$L_{t(0)}/L$ is purely transcendental. Thus
$$\ind (\overline{\varphi} \otimes
L_{t(0)}) = \ind (\overline{\varphi}) = i_0 + \dots + i_{t(0)}.$$
This shows that $\overline{\varphi}_{t(0)}$ is the kernel form of $\overline{\varphi} \otimes
L_{t(0)}$. Suppose now that for some $s \in \{ 1, \dots, e \}$ we
already showed that $\overline{\varphi}_{t(k)}$ is the kernel form
of $\overline{\varphi} \otimes L_{t(k)}$ for $0 \leq k < s$. Then
$(L_{t(k)} \; \vert \; 0 \leq k \leq s)$ is the beginning of a
generic splitting tower of $\overline{\varphi}$. Hence $\overline{\varphi} \otimes
L_{t(s)}$ has index $i_0 +
\dots + i_{t(s)}$, and it follows that $\overline{\varphi}_{t(s)}$
is the kernel form of $\overline{\varphi} \otimes L_{t(s)}$. Thus,
this holds for all $s \in \{ 0, \dots, e \}$. Now it is clear that
$(L_{t(s)}\; \vert \; 0 \leq s \leq e)$ is a generic splitting
tower of $\overline{\varphi}$.\qed
\endproof
