%% Copyright (C) 1990, 1998 by Markus Rost.
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%% "On quadratic forms isotropic over the function field of a conic"
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%% The note appeared in Mathematische Annalen
%% Math. Ann. 288 (1990), no. 3, 511-513.
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\begin{document}

\title{On quadratic forms isotropic over the function field of a
conic}
\author{Markus Rost}
\address{NWF I - Mathematik,
  Universit\"at Regensburg,
  D-93040 Regensburg, Germany}
\email{markus.rost@mathematik.uni-regensburg.de}
\urladdr{http://www.physik.uni-regensburg.de/\~{}rom03516}
\renewcommand{\datename}{}
\date{The text appeared as: \frenchspacing Math. Ann. 288 (1990),
no. 3, 511--513.}
\maketitle

Consider a quadratic extension $L=F(\sqrt a)$ of a field~$F$ ($\Char
F\not= 2$).  The behaviour of quadratic forms over~$F$ under base
extension $\varphi\to\varphi_L$ is well understood, since any
anisotropic form~$\varphi$ is isomorphic to $\psi\perp\langle
1,-a\rangle\varrho$ with forms $\psi$,~$\varrho$ over~$F$ such that
$\psi_L$ is anisotropic~\cite[2.~\S5]{S}.  This implies that the
anisotropic part $(\varphi_L)_\an$ of~$\varphi_L$ is isomorphic
to~$\psi_L$ and therefore already defined over~$F$ and that if
$\varphi$ is anisotropic and $\varphi_L$ is hyperbolic then $\varphi$
is a multiple of $\langle 1,-a\rangle$.

Now let $K$ be the function field of a conic, i.e.\ $K$ is for some
$a$, $b\in F^*$ isomorphic to the fraction field of
$R=F[s,t]/(s^2-at^2-b)$.  Then $K$ is the universal splitting field of
the form $\langle 1,-a,-b\rangle$ and in view of the decomposition
mentioned above it is natural to ask whether a form~$\varphi$ which
becomes isotropic over~$K$ contains a subform similar to $\langle
1,-a,-b\rangle$.  This however is not true in general;
see~\cite[\S6]{L} for further information.  The purpose of this note
is to prove

\begin{proposition}
  Let $\varphi$ be a form over~$F$.  Then there exist a number $p$,
  forms $\varphi_i$, $\psi_i$ \textup($i=0$, \dots, $p$\textup) and
  elements $c_i\in F^*$ \textup($i=0$, \dots, $p-1$\textup) such that
  $\varphi=\varphi_0$ and
  \begin{enumerate}
    \item[i)] $\varphi_i\simeq c_i\langle 1,-a\rangle\perp \psi_i$,
    \quad \text{$i=0$, \dots, $p-1$;}
    \item[ii)] $\varphi_{i+1}\simeq c_ib\langle 1,-a\rangle \perp
    \psi_i$, \quad \text{$i=0$, \dots, $p-1$;}
    \item[iii)] $\bigl((\varphi_p)_K\bigr)_\an \cong
    \bigl((\varphi_p)_\an\bigr)_K$.
  \end{enumerate}
\end{proposition}

This proposition shows that the extension~$K|F$ has similar splitting
properties as the quadratic extension~$L|F$:

\begin{corollary}
  Let $\varphi$ be a form over~$F$.  Then there exists a form~$\psi$
  over~$F$ such that $(\varphi_K)_\an$ is isomorphic to~$\psi_K$.  If
  $\varphi$ is anisotropic and $\varphi_K$ is hyperbolic then
  $\varphi$ is a multiple of $\langle 1,-a,-b,ab\rangle$.
\end{corollary}

The first statement of the corollary has been proved by Arason
in~\cite[Appendix~II]{ELW}; it follows from the proposition by taking
$\psi=(\varphi_p)_\an$, since all~$\varphi_i$ are isomorphic over~$K$.
The second statement is well known, see e.g.~\cite[\S2]{A}
or~\cite[4.~\S5]{S}; it is a consequence of the proposition and Witt
cancellation.  The method of proof presented here is direct and
constructive and might indicate a way to handle the extension~$K|F$
also for other questions.

Note that $R=F[t]\oplus sF[t]$ as $F$-vector space.  Define $d\colon
R\to\IN\cup \{-\infty\}$ by
\begin{equation*}
  d(P+sQ)=\max\{\deg P,1+\deg Q\}\quad\text{for $P$, $Q\in F[t]$}
\end{equation*}
(here $\deg 0= -\infty$).  Moreover let $R_n=\{\,r\in R\mid d(r)\leq
n\,\}$.  $R_n$ is a $F$-vector subspace of~$R$ and one has $R_0=F$ and
$R_n\cdot R_m\subset R_{n+m}$.

\begin{lemma}
  Let $\varphi\colon V\to F$ be an anisotropic form and suppose that
  for some $n\geq 1$ there exist
  \begin{equation*}
    v\in (V\otimes_FR_n)\setminus (V\otimes_FR_{n-1})
  \end{equation*}
  such that $\varphi(v)=0\in R$.

  Then there exists a subspace $L\subset V$ of dimension~$2$ such that
  \begin{enumerate}
    \item[1)] $\varphi|L\simeq c\langle 1,-a\rangle$ for some
    $c\in F^*$,
    \item[2)] there exists a nonzero $\tilde v\in V\otimes_FR_{n-1}$
    such that $\tilde\varphi(\tilde v)=0$, where
    $\tilde\varphi=b(\varphi|L)\perp(\varphi|W)$ and $W=L^\perp $.
  \end{enumerate}
\end{lemma}

\begin{proof}[Proof of the proposition]
  We use induction on $\dim\varphi_\an$.  It is clear that we may
  assume that $\varphi$ is anisotropic and $\varphi_K$ is isotropic.

  Since $K$ is the fraction field of~$R$ there exist $n\geq 0$ and a
  nonzero $v\in V\otimes_FR_n$ such that $\varphi(v)=0$.  We proceed
  by induction on~$n$.  If $n=0$, then $v\in V$ and $\varphi$ would be
  isotropic over~$F$; hence $n\geq 1$.  We may assume $v\not\in
  V\otimes_FR_{n-1}$ and we take $\varphi_1=\tilde\varphi$ where
  $\tilde\varphi$ is the form in the lemma.  If $\tilde\varphi$ is
  anisotropic we apply the induction hypothesis for $n-1$ and if
  $\tilde\varphi$ is isotropic we apply the induction hypothesis for
  $\dim\tilde\varphi_\an< \dim\varphi$.  In any case we find forms
  $\tilde\varphi=\tilde\varphi_0$, $\tilde\varphi_1$,
  \dots,~$\tilde\varphi_p$ as in the proposition and
  $\varphi=\varphi_0$, $\varphi_i=\tilde\varphi_{i-1}$ ($i=1$, \dots,
  $p+1$) is a sequence as required.
\end{proof}

In order to prove the lemma we write
\begin{equation*}
  v=v_0+\sum^n_{i=1} v_i st^{i-1}+w_it^i;\quad \text{$v_i$, $w_i\in
  V$.}
\end{equation*}

\begin{claim}
  $\langle v_n,w_n\rangle_\varphi=0$ and
  $\varphi(w_n)=-a\varphi(v_n)$.
\end{claim}

\begin{proof}[Proof of the claim:]
  \begin{align*}
    0 &= \varphi(v)\bmod R_{2n-1}\\
    &= \varphi(v_n)s^2t^{2(n-1)}+2\langle v_n,w_n\rangle_\varphi
    st^{2n-1}+\varphi(w_n)t^{2n}\bmod R_{2n-1}\\
    &= (\varphi(v_n)a+\varphi(w_n))t^{2n}+2\langle
    v_n,w_n\rangle_\varphi st^{2n-1}\bmod R_{2n-1}.
  \end{align*}
  The claim follows since $t^{2n}$ and~$st^{2n-1}$ define
  $F$-independent vectors of~$R/R_{2n-1}$.  \noqed
\end{proof}

Note that $v_n\not= 0$ and $w_n\not= 0$ since $v\not\in
V\otimes_FR_{n-1}$ and $\varphi$ is anisotropic.  Let
\begin{equation*}
  L=F[z]/(z^2-a)
\end{equation*}
and let $\alpha\in L^*$ be the class of~$z$.  We identify $L$ with
$\langle v_n,w_n\rangle_F\subset V$ by $1\to v_n$ and $\alpha\to w_n$.
Then the claim shows that $\varphi|L=c N_{L|F}$ with $c=\varphi(v_n)$
and $N_{L|F}\colon L\to F$, $e+\alpha f\to e^2-af^2$ the norm form.

Now write $v=x+y$ with $x\in L\otimes_FR$ and $y\in W\otimes_FR$,
$W=L^\perp$.  Then $x\in (s+t\alpha)t^{n-1} +L \otimes_FR_{n-1}$ and
$y\in W\otimes_FR_{n-1}$.

Put $\tilde v=b^{-1}(s-t\alpha)x+y$.  Then $\tilde v$ is a zero of the
form $\tilde\varphi=b(\varphi|L)\perp (\varphi|W)$, since
\begin{align*}
  b\varphi(b^{-1}(s-t\alpha)x)&=bc N_{L|F}(b^{-1}(s-t\alpha)x)=bc
  b^{-2}(s^2-at^2)N_{L|F}(x)\\
  &=c N_{L|F}(x)=\varphi(x).
\end{align*}
It remains to show that $\tilde v\in V\otimes_FR_{n-1}$.  In order to
do this we have to show that $(s-t\alpha)x\in L\otimes_FR_{n-1}$.

{\it Case I: $n\geq 2$.} Then there exist $\mu$, $\lambda\in L$ and
$\tilde x\in L\otimes_FR_{n-2}$ such that
\begin{equation*}
  x=(s+t\alpha)t^{n-1}+(s+t\alpha)t^{n-2}\mu+t^{n-1}\lambda+\tilde x
\end{equation*}
We have with $\omega=t^{n-1}+t^{n-2}\mu$ and $\bar\lambda$ the
conjugate of $\lambda$ under $\alpha\to -\alpha$:
\begin{align*}
  0 &= \varphi(v)\bmod R_{2n-2}=\varphi(x)\bmod R_{2n-2}\\
  &= c N_{L|F}((s+t\alpha)\omega+t^{n-1}\lambda)\bmod
  R_{2n-2}\\
  &= c [N_{L|F}((s+t\alpha)\omega)+\tr_{L|F}((s+t\alpha)\omega
  t^{n-1}\bar\lambda)+N_{L|F}(t^{n-1}\lambda)]\bmod R_{2n-2}\\
  &= c [b\cdot
  N_{L|F}(\omega)+\tr_{L|F}((s+t\alpha)t^{2n-2}\bar\lambda)+0]\bmod
  R_{
  2n-2}\\
  &= c [0+st^{2n-2} \tr_{L|F}\bar\lambda
  +t^{2n-1}\tr_{L|F}(\alpha\bar\lambda)]\bmod R_{2n-2}
\end{align*}
Hence the traces of $\bar\lambda$ and $\alpha\bar\lambda$ are zero and
therefore $\lambda =0$.  Finally:
\begin{equation*}
  (s-t\alpha) x=b\omega +(s-t\alpha)\tilde x\in L\otimes_FR_{n-1}.
\end{equation*}

{\it Case II: $n=1$.} Then $x=s+t\alpha+\lambda$ for some $\lambda\in
L$ and it suffices to show $\lambda =0$.  However
\begin{equation*}
  0=\varphi(v)=b+s \tr_{L|F}\bar\lambda+t\; \tr_{L|F}\alpha\bar\lambda
  +N_{L|F}(\lambda)+\varphi(y)
\end{equation*}
and therefore again $\lambda =0$ since $\varphi(y)\in F$.\qed

\begin{thebibliography}{ELW}

  \bibitem[A]{A} Arason, J. Kr.: Cohomologische Invarianten
  quadratischer Formen. J. Algebra {\bf 36}, 448--491 (1975).

  \bibitem[ELW]{ELW} Elman, R., Lam, T. Y., Wadsworth, A. R.: Amenable
  fields and Pfister extensions. Conference on quadratic forms 1976,
  Queen's pap. Pure Appl. Math. {\bf 46}, 445--491 (1977).

  \bibitem[L]{L} Lam, T. Y.: Fields of $u$-invariant~$6$ after A.
  Merkuriev. In: ``Ring theory 1989'' in honour of S. A. Amitsur, Isr.
  Conf. Proc., vol. 1, pp. 12--30.

  \bibitem[S]{S} Scharlau, W.: Quadratic and Hermitian forms.
  Berlin-Heidelberg-New York: Springer 1985.

\end{thebibliography}
\end{document}

