%% Copyright (C) 1999, 2024 Markus Rost.
%%
%% This file contains the TeX source code for the article
%% "A descent property for Pfister forms".
%%
%% This text is a simplified and extended version of a preprint dated
%% from June 1991.
%%
%% It appeared in the Journal of the Ramanujan Mathematical Society:
%% J. Ramanujan Math. Soc. 14 No. 1 (1999) pp. 55-63.
%%
%% The output from this file may differ slightly from the article.
%%
%% AMS-LaTeX file.
%% To be formatted (twice) with LaTeX2e.
%% 6 pages and 26kb of output.
%% Compilation should work at least with versions LaTeX2e <1997/12/01>
%% and documentclass amsart 1997/03/26 v1.2r or later.
%%
%% Last Changes:
%% Jan  6 1999 (ready for submission)
%% Jun  8 1999 (the proofs were OK.  No changes)
%% Aug 12 1999 (it appeared!)
%% Sep  5 1999 Misprint: \section{Descent for extensions of odd degree}
%% Oct  9 1999 (Clean up the stupid proof of the lemma)
%% Dec 11 2024 Undo the Sep 5 1999 change.  Changes are marked with "%% Dec 2024"

\NeedsTeXFormat{LaTeX2e}
\documentclass{amsart}

\DeclareSymbolFont{Fraktur}{U}{euf}{b}{n}
\DeclareMathSymbol{\dm}{\mathalpha}{Fraktur}{"6D}
\DeclareMathOperator{\Br}{Br}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\car}{char}
\let\tensor\otimes
\newcommand{\LZ}{\mathbf Z}
\newcommand{\amod}{{\rm mod}\mkern5mu} % \bmod with no left argument
\newcommand{\id}{\mathrm{id}}
\newcommand{\inv}{^{-1}}
\newcommand{\orth}{\mathbin\bot}
\newcommand{\pfister}[1]{\langle\!\langle#1\rangle\!\rangle}
\newcommand{\qform}[1]{\langle#1\rangle}

\theoremstyle{plain}
\newtheorem*{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{proposition}{Proposition}

\begin{document}

\title{A descent property for Pfister forms}
\author{Markus Rost}
\address{NWF I - Mathematik,
  Universit\"at Regensburg,
  D-93040 Regensburg, Germany}
%\curraddr{}
\email{markus.rost@mathematik.uni-regensburg.de}
\urladdr{http://www.physik.uni-regensburg.de/\~{}rom03516}
%\thanks{}
%\date{\today}
% \date{January 6, 1998}
%% Dec 2024
\date{June 1991; January 1998; October 1999; December 11, 2024}
\maketitle

\section*{Summary}

The Rosenberg-Ware theorem states that for a Galois extension~$K/F$ of
odd degree the natural map of Witt rings of quadratic forms
\begin{equation*}
  W(F)\to W(K)^{\Gal(K/F)}
\end{equation*}
is an isomorphism.  We extend this result to arbitrary field
extensions~$K/F$ of odd degree.  Basically we show that
(Proposition~\ref{prop:1})
\begin{equation*}
  0\to W(F)\xrightarrow{r_{K/F}}W(K)\xrightarrow{i_1-i_2}W(K\tensor K)
\end{equation*}
is exact, where $i_1$,~$i_2$ are induced from the two natural maps
$K\to K\tensor K$.  Further it is shown that an element of the graded
Witt ring is represented by a Pfister form if this is true after an
extension of odd degree (Proposition~\ref{prop:3}).  We apply this to
trace forms of exceptional Jordan algebras (Proposition~\ref{prop:4}).
In the last section similar questions for symbols in Milnor's
$K$-theory and Galois cohomology are considered.

The final version of this article has been prepared during a visit at
ETH Zurich.  I thank the FIM at ETH Zurich for its hospitality.

\section{The transfer map}

For generalities on quadratic forms we refer to
\cite{Lam:73,Scharlau:85}.

Let $F$ be a field of characteristic different from~$2$ and let $W(F)$
be the Witt group of quadratic forms over~$F$.  Due to the Witt
cancelation theorem one may identify $W(F)$ with the set of
isomorphism classes of anisotropic quadratic forms over~$F$.

For a field extension~$K/F$ we denote by
\begin{gather*}
  r_{K/F}\colon W(F)\to W(K),\\
  r_{K/F}([\varphi])=[\varphi_K]
\end{gather*}
the homomorphism given by extension of scalars.

Let $s\colon K\to F$ be a nontrivial $F$-linear map.  According to
Scharlau~\cite[Chap.~2, \S~5]{Scharlau:85} there is an associated
transfer map
\begin{gather*}
  s_*\colon W(K)\to W(F),\\
  s_*([\psi])=[s\circ\psi]
\end{gather*}
such that
\begin{equation*}
  s_*\circ r_{K/F}(x)=s_*(1)x.
\end{equation*}

%% Dec 2024: keep the former page break
\goodbreak

If $K/F$ has odd degree, then $s$ can be chosen so that
\begin{equation*}
  s_*(1)=1,
\end{equation*}
cf.~\cite[Chap.~2, Lemma~5.8]{Scharlau:85}.  In this case $s_*$ is a
left inverse to~$r_{K/F}$.  It follows that the restriction
map~$r_{K/F}$ is injective for extensions~$K/F$ of odd degree.

If $K/F$ is purely inseparable (necessarily of odd degree since $\car
F\neq 2$), then $K$ and~$F$ have the same square class groups.  Hence
every quadratic form over~$K$ is extended from a form over~$F$, as can
be seen via diagonalization.  Therefore $W(F)=W(K)$ in the purely
inseparable case.

Let $I(F)\subset W(F)$ be the fundamental ideal consisting of the
classes of even dimensional quadratic forms and let $I^n(F)$ denote
its $n$-th power.  The transfer maps respect the filtration of the
Witt ring by the powers of its fundamental ideal, i.~e.,
\begin{equation*}
  s_*\bigl(I^n(K)\bigr)\subset I^n(F),
\end{equation*}
cf.~\cite[Lemma 3.2]{Arason:75}.  It follows that
\begin{equation*}
  I^n(F)=r_{K/F}\inv\bigl(I^n(K)\bigr)
\end{equation*}
for an extension~$K/F$ of odd degree.

% \section{Descent for extension of odd degree}
%% Dec 2024:
\section{Descent for extensions of odd degree}

Let $K/F$ be a finite field extension.  For each prime ideal~$\dm$ of
$K\tensor_FK$ let
\begin{equation*}
  H_\dm=(K\tensor_FK)/\dm,
\end{equation*}
and let
\begin{equation*}
  i_\dm^1,\,i_\dm^2\colon W(K)\to W(H_\dm)
\end{equation*}
be the restriction maps induced from the homomorphisms
\begin{equation*}
  K\to H_\dm,\quad a\mapsto a\tensor 1\bmod\dm,\quad a\mapsto 1\tensor
  a\bmod\dm,
\end{equation*}
respectively.  We put
\begin{gather*}
  \delta\colon W(K)\to\prod_\dm W(H_\dm),\\
  \delta(x)=\bigl((i_\dm^1-i_\dm^2)(x)\bigr)_\dm.
\end{gather*}

\begin{proposition}
  \label{prop:1}
  Let $K/F$ be an extension of odd degree.  Then the sequence
  \begin{equation*}
    0 \to W(F) \xrightarrow{r_{K/F}} W(K) \xrightarrow{\delta\,}
    \prod_\dm W(H_\dm)
  \end{equation*}
  is exact.
\end{proposition}

\begin{proof}
  The injectivity of~$r_{K/F}$ has been discussed already.  Further,
  using the bijectivity of~$r_{K/F}$ in the purely inseparable case,
  one easily reduces to separable extensions~$K/F$.  In this case one
  has
  \begin{equation*}
    K\tensor_FK=\bigoplus_\dm H_\dm.
  \end{equation*}

  Let $s\colon K\to F$ be $F$-linear with $s_*(1)=1$ and let
  $s_\dm\colon H_\dm\to K$ be the components of $s\tensor \id_K$,
  i.~e.,
  \begin{equation*}
    s(a)b=\sum_\dm s_\dm(a\otimes b\bmod \dm).
  \end{equation*}
  With these settings one has for $x\in W(K)$
  \begin{align*}
    \sum_\dm (s_\dm)_*\bigl(i_\dm^1(x)\bigr)&=r_{K/F}\circ s_*(x),\\
    \sum_\dm (s_\dm)_*\bigl(i_\dm^2(x)\bigr)&=s_*(1)x=x,
  \end{align*}
  as may be verified on the level of forms.

  If $x\in \ker\delta$, then $i_\dm^1(x)=i_\dm^2(x)$ for all~$\dm$,
  whence $x=r_{K/F}\circ s_*(x)$.
\end{proof}

The Rosenberg-Ware theorem appears as a special case of
Proposition~\ref{prop:1}.  Namely if $K/F$ is a Galois extension with
Galois group~$G$, the map~$\delta$ can be identified with the
homomorphism
\begin{gather*}
  W(K)\to\prod_{\sigma\in G} W(K),\\
  x\mapsto\bigl((1-\sigma)(x)\bigr)_\sigma.
\end{gather*}
Therefore $\ker\delta$ equals the subgroup of Galois invariants.

\section{Descent of Pfister forms}

An $n$-fold Pfister form is a quadratic form of type
\begin{equation*}
  \pfister{a_1,\dots,a_n}=\bigotimes_{i=1}^n\qform{1,-a_i}
\end{equation*}
with $a_i\in F^*$.

The classes of $n$-fold Pfister forms generate~$I^n(F)$.  If a Pfister
form is isotropic, then it is hyperbolic~\cite[Chap.~Ten]{Lam:73}.

A basic theorem of Pfister \cite[Chap.~4, Theorem~4.4]{Scharlau:85}
asserts that an anisotropic quadratic form~$\varphi$ in indeterminates
$X=(x_1,\dots,x_m)$ is isomorphic to a Pfister form if and only if
\begin{equation}
  \label{eq:1}
  \varphi(X)\varphi_{F(X)}\simeq\varphi_{F(X)}.
\end{equation}

If $\alpha$ and~$\beta$ are $n$-fold Pfister forms such that
\begin{equation*}
  \alpha=\beta\bmod I^{n+1}(F)
\end{equation*}
in $W(F)/I^{n+1}(F)$, then $\alpha$ and~$\beta$ are isomorphic
\cite[Chap.~Ten, Corollary~3.4]{Lam:73}.

\begin{lemma}
  Let\/ $\varphi$ be a quadratic form over~$F$, let $K/F$ be an
  extension of odd degree, and suppose that $\varphi_K$ is isomorphic
  to a Pfister form.  Then $\varphi$ is isomorphic to a Pfister form.
\end{lemma}

\begin{proof}
%%%%% Dec 2024:  Add the first email from Pfister
%% From: "Prof. Pfister" <PFISTER@Mathematik.Uni-Mainz.DE>
%% Organization:  Department of Mathematics
%% To: markus.rost@mathematik.uni-regensburg.de
%% Date:          Fri, 1 Oct 1999 11:50:43 +0200 (MESZ)
%% Subject:       
%% Priority: normal
%% 
%% Lieber Herr Rost,
%% vielen Dank fuer Ihren Sonderdruck, den ich gleich gelesen habe (der 
%% Beweis von  Lemma 1 ist etwas missglueckt).
%% 
%% ...
%% 
%% Herzlichen Dank und alle guten Wuensche fuer Princeton,
%% Ihr A. Pfister
%%%%%
%% From: ``Prof. Pfister'' <PFISTER@Mathematik.Uni-Mainz.DE>
%% Organization:  Department of Mathematics
%% To: Markus  Rost <rost@IAS.EDU>
%% Date:          Tue, 5 Oct 1999 12:49:05 +0200 (MESZ)
%% Subject:       Re:        
%% Priority: normal
%
%% Lieber Herr Rost,
%% 
%% ...
%% 
%% Zu Ihrem Lemma 1 ist zu sagen:
%% Im isotropen Fall (ueber K) ist phi hyperbolisch und von Zweipotenz-
%% Dimension, also eine Pfister Form.
%% Im anisotropen Fall ist die Aussage ``phi anisotrop'' trivial, der
%% Rest 
%% folgt mit (1).
%% Viele Gruesse
%% Ihr A.Pfister
%
%%   If $\varphi_K$ is isotropic, it is hyperbolic, and therefore
%%   $\varphi$ is hyperbolic.
%%
%%   Assume that $\varphi_K$ is anisotropic.  Then $\varphi$ is
%%   anisotropic by Springer's theorem~\cite[Chap.~2,
%%   Theorem~5.3]{Scharlau:85}.  The claim follows from the
%%   criterion~\eqref{eq:1} and the injectivity of $W\bigl(F(X)\bigr)\to
%%   W\bigl(K(X)\bigr)$.
  Since $\varphi_K$ is a Pfister form, the dimension of~$\varphi$ is a
  $2$-power.  If $\varphi_K$ is isotropic, it is hyperbolic and
  therefore $\varphi$ is hyperbolic.  It follows that $\varphi$ is a
  Pfister form.

  Assume that $\varphi_K$ is anisotropic.  Then $\varphi$ is
  anisotropic and the claim follows from the criterion~\eqref{eq:1}
  and the injectivity of $W\bigl(F(X)\bigr)\to W\bigl(K(X)\bigr)$.
\end{proof}

\begin{proposition}
  \label{prop:3}
  Let\/ $\varphi$ be a quadratic form over~$F$, let $K/F$ be an
  extension of odd degree, and suppose that there exists an $n$-fold
  Pfister form~$\beta$ over~$K$ such that
  \begin{equation*}
    \varphi_K=\beta\bmod I^{n+1}(K).
  \end{equation*}
  Then there exists an $n$-fold Pfister form~$\alpha$ over~$F$ such
  that
  \begin{equation*}
    \varphi=\alpha\bmod I^{n+1}(F).
  \end{equation*}
\end{proposition}

\begin{proof}
  Let $H/F$ be a field extension and let $f$, $g\colon K\to H$ be two
  homomorphisms over~$F$.  We denote the by $r_f$ resp.~$r_g$ the
  extension of scalars via $f$ resp.~$g$.  Obviously
  \begin{equation*}
    r_f(\beta)=r_g(\beta) \bmod I^{n+1}(H).
  \end{equation*}
  Hence $r_f(\beta)=r_g(\beta)$ in~$W(H)$.

  Using the last equality for the fields~$H_\dm$, it follows from
  Proposition~\ref{prop:1} that $\beta\in r_{K/F}\bigl(W(F)\bigr)$.
  The Lemma shows $\beta=\alpha_K$ for some Pfister form~$\alpha$.
  Then $(\varphi-\alpha)_K\in I^{n+1}(K)$ and therefore
  $\varphi-\alpha\in I^{n+1}(F)$.
\end{proof}

\begin{corollary}
  \label{cor:1}
  Let\/ $0\leq n_1<n_2<\dots<n_r$ be integers and let $c_1$, \dots,
  $c_r\in F^*$.  Let further $\varphi$ be a quadratic form over~$F$,
  let $K/F$ be an extension of odd degree, and suppose that there
  exist $n_i$-fold Pfister forms~$\beta_i$ over~$K$ \textup($i=1$,
  \dots,~$r$\textup) such that
  \begin{equation*}
    \varphi_K=c_1\beta_1+c_2\beta_2+\dots+c_r\beta_r
  \end{equation*}
  in~$W(K)$.  Then there exist $n_i$-fold Pfister forms~$\alpha_i$
  over~$F$ such that $\beta_i=(\alpha_i)_K$ \textup($i=1$,
  \dots,~$r$\textup) and
  \begin{equation*}
    \varphi=c_1\alpha_1+c_2\alpha_2+\dots+c_r\alpha_r
  \end{equation*}
  in~$W(F)$.  Moreover, if for some $j>i$ one has
  $\beta_j=\pfister{b_1,\dots,b_s}\beta_i$ for some $b_k\in K^*$, then
  $\alpha_j=\pfister{a_1,\dots,a_s}\alpha_i$ for some $a_k\in F^*$.
\end{corollary}

\begin{proof}
  For the first statement we use induction on~$r\geq 0$.  One has
  $\varphi_K=\beta_1\bmod I^{n_1+1}(K)$.  By Proposition~\ref{prop:3}
  there exists an $n_1$-fold Pfister form~$\alpha$ over~$F$ such that
  $\varphi=\alpha\bmod I^{n_1+1}(F)$.  Then necessarily
  $\beta_1=\alpha_K$.  The claim follows by applying the induction
  hypothesis to the form~$\varphi\orth-c_1\alpha$.

  For the second statement first note that $\alpha_i$ is a subform
%%  of~$\alpha_j$ by Springer's theorem.  The claim follows
  of~$\alpha_j$ by Springer's theorem~\cite[Chap.~2,
  Theorem~5.3]{Scharlau:85}.  The claim follows
  from~\cite[Theorem~2.7]{Elman-Lam:72}.
\end{proof}

Quadratic forms~$\varphi$ of the type as in Corollary~\ref{cor:1}
appear when studying trace forms of various algebras.  An example is
considered in the next section.

\section{Serre's $(\amod 2)$ invariants for~$F_4$}

It has been noticed by Serre that there are cohomological invariants
\begin{align*}
  f_3\colon H^1(F,F_4)\to H^3(F,\LZ/2),\\
  f_5\colon H^1(F,F_4)\to H^5(F,\LZ/2),
\end{align*}
cf.~\cite[III.~Annexe, \S~3.4]{Serre:94} or~\cite[III.~Appendix~2,
3.4]{Serre:97} and~\cite[\S~40]{Knus-Merkurjev-Rost-Tignol:98}.  The
construction of these invariants is based on the interpretation of
$H^1(F,F_4)$ as the set of isomorphism classes of exceptional Jordan
algebras
(cf.~\cite{Jacobson:68,Knus-Merkurjev-Rost-Tignol:98,Springer:98}) and
the following description of their trace forms:
\begin{proposition}
  \label{prop:4}
  Let $J$ be an exceptional Jordan algebra over~$F$ and let
  \begin{gather*}
    q_J\colon J\to F,\\
    q_J(x)=T_J(x^2),
  \end{gather*}
  where $T_J$ denotes the trace map of~$J$.  Then there exist elements
  $a_1$, \dots, $a_5\in F^*$ such that
  \begin{equation}
    \label{eq:2}
    q_J\simeq \qform{1,1,1}\orth
    2\pfister{a_1,a_2,a_3}\qform{-a_4,-a_5,a_4a_5}.
  \end{equation}
\end{proposition}
In terms of this description, Serre's invariants are given by
\begin{equation*}
  f_3([J])=(a_1)(a_2)(a_3), \quad f_5([J])=(a_1)(a_2)(a_3)(a_4)(a_5).
\end{equation*}

For a proof of Proposition~\ref{prop:4} using an analysis of the Tits
constructions of exceptional Jordan algebras see~\cite[Lemma
40.1]{Knus-Merkurjev-Rost-Tignol:98}.

Alternatively one can prove Proposition~\ref{prop:4} using
Corollary~\ref{cor:1} as follows.  Let $\varphi=2\bigl(q_J\orth
-\qform{1,1,1}\bigr)$.  After passing to an appropriate cube
extension~$K/F$, the Jordan algebra~$J$ has zero divisors.  In this
case (the so called ``reduced'' case), the description~\eqref{eq:2} of
the trace form can be read off a presentation of~$J$ in terms of
$3\times 3$ matrices over an octonion
algebra~\cite{Jacobson:68,Knus-Merkurjev-Rost-Tignol:98}.  Hence
$\varphi_K=\pfister{b_1,b_2,b_3,b_4,b_5}-\pfister{b_1,b_2,b_3}$
in~$W(K)$ for some~$b_i\in K^*$.  By Corollary~\ref{cor:1} one has
$\varphi=\pfister{a_1,a_2,a_3,a_4,a_5}-\pfister{a_1,a_2,a_3}$
in~$W(F)$ for some~$a_i\in F^*$, whence~\eqref{eq:2}.

\section{Descent for $(\amod 2)$ symbols in Milnor's $K$-ring}

Let $K^M_n F$ be Milnor's $K$-group
of~$F$~\cite{Bass-Tate:72,Milnor:70}.  For $a_1$, \dots, $a_n\in F^*$
one denotes by $\{a_1,\dots,a_n\}$ the image of $a_1\tensor\dots
\otimes a_n$ in~$K^M_n F$.

Let $p\neq\car F$ be a prime.  By a \emph{$(\amod p)$-symbol} we
understand an element in $K^M_n F/p$ of the form
$\{a_1,\dots,a_n\}\bmod pK^M_n F$.  Let us call an element $x\in K^M_n
F/p$ a \emph{weak $(\amod p)$-symbol}, if there exists a finite field
extension~$K/F$ of degree prime to~$p$ such that $x_K$ is a $(\amod
p)$-symbol.

Is a weak $(\amod p)$-symbol always a $(\amod p)$-symbol?

I don't know any counterexample.  For $p=2$ one has:

\begin{corollary}
  \label{cor:weak}
  Every weak $(\amod 2)$-symbol is a $(\amod 2)$-symbol.
\end{corollary}

\begin{proof}
  Milnor~\cite{Milnor:70} defined a homomorphism
  \begin{gather*}
    s_n\colon K^M_nF/2\to I^n(F)/I^{n+1}(F),\\
    s_n(\{a_1,\dots,a_n\})=\pfister{a_1,\dots,a_n}\bmod I^{n+1}(F).
  \end{gather*}
  The map~$s_n$ is surjective and it is injective on
  symbols~\cite[Prop.~2.1]{Elman-Lam:72}.

  Let $x\in K^M_nF/2$, let $K/F$ be of odd degree, and suppose that
  \begin{equation*}
    x_K=\{b_1,\dots ,b_n\}
  \end{equation*}
  with $b_i\in K^*$.  Let $\varphi$ be a quadratic form such that
  \begin{equation*}
    s_n(x)=\varphi\bmod I^{n+1}(F).
  \end{equation*}
  Applying Milnor's homomorphism over~$K$ yields
  \begin{equation*}
    \pfister{b_1,\dots,b_n}=\varphi_K\bmod I^{n+1}(K).
  \end{equation*}
  By Proposition~\ref{prop:3} there exist $a_i\in F^*$ with
  \begin{equation*}
    s_n(x)=s_n(\{a_1,\dots ,a_n\}).
  \end{equation*}
  Since $s_n$ is injective on symbols one has
  \begin{equation*}
    x_K=\{a_1,\dots ,a_n\}_K\bmod 2K^M_nK.
  \end{equation*}
  Applying the transfer map in Milnor's $K$-theory yields
  \begin{equation*}
    x=\{a_1,\dots ,a_n\}\bmod 2K^M_nF
  \end{equation*}
  since $K/F$ is of odd degree.
\end{proof}

If $n=2$ and $\mu_p\subset F$, the question whether a weak symbol is a
symbol is equivalent to the question whether an algebra of prime
degree~$p$ is cyclic.  This is known for $p\leq 3$ and unsettled
otherwise.

Using the results of~\cite{Haile-Knus-Rost-Tignol:96} one can show
that every weak symbol in $K^M_2F/3$ is a symbol for any field~$F$ of
characteristic different from $2$ and~$3$.

Beyond the cases $p=2$ and $n=2$, $p=3$ not much seems to be known,
even for instance for the following question: Let $K/F$ be a quadratic
extension, let $x\in K^M_3F/3$ and suppose that $x_K$ is a symbol.  Is
then $x$ itself a symbol?  Similar for $x\in K^M_2F/5$.

% \bibliography{general}
% \bibliographystyle{amsplainR}
% \end{document}

\raggedbottom \providecommand{\mhy}{-}
\providecommand{\transl}[1]{[#1]}

\begin{thebibliography}{10}

  \bibitem{Arason:75} J.~K. Arason, \emph{Cohomologische Invarianten
  quadratischer Formen}, J.  Algebra \textbf{36} (1975), no.~3,
  448--491.

  \bibitem{Bass-Tate:72} H.~Bass and J.~Tate, \emph{The Milnor ring of
  a global field}, Algebraic $K$-theory, II: ``Classical'' algebraic
  $K$-theory and connections with arithmetic (Proc. Conf., Seattle,
  Wash., Battelle Memorial Inst., 1972) (H.~Bass, ed.), Lecture Notes
  in Mathematics, vol. 342, Springer, Berlin, 1973, pp.~349--446.

  \bibitem{Elman-Lam:72} R.~Elman and T.~Y. Lam, \emph{Pfister forms
  and $K$-theory of fields}, J.  Algebra \textbf{23} (1972), 181--213.

  \bibitem{Haile-Knus-Rost-Tignol:96} D.~E. Haile, M.-A. Knus,
  M.~Rost, and J.-P. Tignol, \emph{Algebras of odd degree with
  involution, trace forms and dihedral extensions}, Israel J. Math.
  \textbf{96} (1996), part B, 299--340.

  \bibitem{Jacobson:68} N.~Jacobson, \emph{Structure and
  representations of Jordan algebras}, American Mathematical Society,
  Providence, R.I., 1968, American Mathematical Society Colloquium
  Publications, Vol. XXXIX.

  \bibitem{Knus-Merkurjev-Rost-Tignol:98} M.-A. Knus, A.~Merkurjev,
  M.~Rost, and J.-P. Tignol, \emph{The book of involutions}, American
  Mathematical Society Colloquium Publications, vol.~44, American
  Mathematical Society, Providence, RI, 1998, With a preface in French
  by J.\ Tits.

  \bibitem{Lam:73} T.~Y. Lam, \emph{The algebraic theory of quadratic
  forms}, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program,
  Reading, Mass., 1980, Revised second printing, Mathematics Lecture
  Note Series.

  \bibitem{Milnor:70} J.~Milnor, \emph{Algebraic $K$-theory and
  quadratic forms}, Invent. Math.  \textbf{9} (1969/1970), 318--344.

  \bibitem{Scharlau:85} W.~Scharlau, \emph{Quadratic and Hermitian
  forms}, Grundlehren der mathematischen Wissenschaften, vol. 270,
  Springer-Verlag, Berlin, 1985.

  \bibitem{Serre:94} J.-P. Serre, \emph{Cohomologie galoisienne},
  fifth ed., Lecture Notes in Mathematics, vol.~5, Springer-Verlag,
  Berlin, 1994.

  \bibitem{Serre:97} \bysame, \emph{Galois cohomology},
  Springer-Verlag, Berlin, 1997, Translated from the French by Patrick
  Ion and revised by the author.

  \bibitem{Springer:98} T.~A. Springer, \emph{Jordan algebras and
  algebraic groups}, Springer-Verlag, Berlin, 1998, Reprint of the
  1973 edition.

\end{thebibliography}

\end{document}
