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\author{\sc{Nicolas Grenier-Boley}, \sc{Detlev W. Hoffmann}\\[2ex]
With an appendix by {\sc Claus Scheiderer}}

\title{Isomorphism criteria for Witt rings of real fields}
%\vspace*{0.3cm} with an appendix by Claus Scheiderer}

\date{}

\begin{document}

\maketitle \noindent \textbf{Abstract.} We prove 
isomorphism criteria for Witt rings and reduced Witt rings of 
certain types of real fields.  Refined criteria are obtained under
the additional assumption that the field be SAP. This leads to
a generalization of a result by Koprowski on Witt equivalence of
function fields of transcendence degree $1$ over a real closed
field. Isomorphism
criteria are also obtained for Witt rings of hermitian forms
over a quadratic extension of a real base field and for Witt groups
of hermitian forms over a quaternion algebra with a real field as center. 
All these criteria are expressed in terms of
properties involving topological subspaces of the space of
orderings of the base field.\\

\noindent\textbf{Mathematics subject classification (2000):} Primary
11E10 ; Secondary 11E04, 11E81, 12D15\\

\noindent\textbf{Keywords:} Witt ring, reduced Witt ring, Witt group, real
field, ordering, preordering, Witt equivalence, hermitian form, quadratic form,
quaternion algebra, function field.

\section{Introduction}

\indent

Throughout, we will only consider fields of characteristic different
from $2$. In \cite{harrison}, Harrison gave necessary and sufficient
conditions for two fields to have isomorphic Witt rings:
\begin{thm}[Harrison]\label{harriso} Let $K$ and $L$ be two fields. Then
the following are equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W(K) \simeq W(L)$.
\item[(2)] There is a group isomorphism $t: {K}^ {*}/{{K}^{*}}^{2} \to
{L}^{*}/{{L}^{*}}^{2}$ sending $-1$ to $-1$ such that the quadratic
form $\l 1,-x,-y,xy \r$ is hyperbolic over $K$ if and only if the
quadratic form $\l 1,-t(x), -t(y), t(x)t(y)\r$ is hyperbolic over
$L$ for all $x, \ y \in {K}^*$.
\end{enumerate}
\end{thm}\noindent
Over global fields, in \cite{pscl}, Perlis, Szymiczek, Conner and
Litherland showed that the previous criterion can be expressed in
terms of (nontrivial) places of the  considered fields:
\begin{thm}[Perlis, Szymiczek, Conner, Litherland]\label{reci}
Let $K$ and $L$ be two global fields and denote by $\Omega_K$ (resp.
$\Omega_L$) the set of nontrivial
places over $K$ (resp. $L$). Then the following are equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W(K) \simeq W(L)$.
\item[(2)] There is a pair of maps $(t,T)$ with a group
isomorphism $t :{K}^*/{{K}^*}^2\to {L}^*/{{L}^*}^2$ and a
bijection $T :\Omega_{K} \to \Omega_{L}$ such that we have the
following equality of Hilbert symbols
$$(x,y)_P=(tx,ty)_{T(P)}$$
for all $x, \ y \in {K}^*/{{K}^*}^2$ and for all $P \in \Omega_{K}$.
\end{enumerate}
\end{thm}

If $K$ is a field, denote by $X_K$ the topological space of all
orderings in $K$. The purpose of this paper is to prove isomorphism
criteria for several types of Witt rings or Witt groups over real base fields. 
It turns out that our results
can be stated in the same way as Theorem \ref{reci} by replacing the
set of nontrivial places by certain subspaces of $X_K$ and local conditions on Hilbert
symbols by local positivity conditions.

Before we state our first main result in the case of Witt rings of real fields,
let us recall that the $u$-invariant $u(K)$ of a field is defined as
$$u(K)=\sup\{\dim q\,|\,\mbox{$q$ is an anisotropic torsion quadratic 
form over $K$}\}\ ,$$
where $q$ is said to be torsion if its Witt class is a torsion element in $W(K)$. 
\begin{thm}\label{wittiso}
Let $K$ and $L$ be two real fields.  Consider the following three
statements: 
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W(K) \simeq W(L)$.
\item[(2)] There is a pair of maps $(t,T)$ with a group
isomorphism $t :{K}^*/{{K}^*}^2\to {L}^*/{{L}^*}^2$ and a homeomorphism
$T: X_K \to X_L$ such that $x$ is positive at $P$ if and only if 
$t(x)$ is positive at $T(P)$,
for all $x \in {K}^*/{{K}^*}^2$ and for all $P \in X_K$.
\item[(3)] There is a pair of maps $(t,T)$ with a group
isomorphism $t :{K}^*/{{K}^*}^2\to {L}^*/{{L}^*}^2$
and a bijection $T: X_K \to X_L$ such that $x$ is
positive at $P$ if and only if $t(x)$ is positive at $T(P)$, for all
$x \in {K}^*/{{K}^*}^2$ and for all $P \in X_K$.
\end{enumerate}
Then $(1)\Longrightarrow (2) \Longleftrightarrow (3)$.  If, in addition,
$u(K)$, $u(L)\leq 2$, then these statements are equivalent.
\end{thm}\noindent
A crucial ingredient in the proof is another result
due to Harrison which asserts that there is a bijection between the
ideals of characteristic $0$ in the spectrum of the Witt ring of a
field and the orderings of that field. This result and several other
facts and notations are recalled in the first part of Section
\ref{sec2}, the second part being dedicated to the proof of Theorem
\ref{wittiso}. 

In Section \ref{sec3}, we further assume that the base fields are SAP fields, 
i.e. that they have the strong approximation property, see Definition \ref{SAP-def}.
Our second main result is the following refinement of Theorem \ref{wittiso}:
\begin{thm}\label{wittsap}
Let $K$ and $L$ be two SAP fields such that $u(K), \ u(L) \leq 2$. Then 
the following are equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W(K)\simeq W(L)$.
\item[(2)] There is a homeomorphism $X_K \simeq X_L$ and a group isomorphism
$\dot{\sigma}(K)/{K^*}^2\\ \simeq \dot{\sigma}(L)/{L^*}^2$.
\end{enumerate}
\end{thm}\noindent
Here, and in the sequel, $\sigma(K)$ stands for the set of elements of $K$
that can be expressed as a sum of squares in $K$, and we write $\dot{\sigma}(K)$ 
for $\sigma (K) \setminus \lbrace 0 \rbrace$ (which is a subgroup of $K^*$).
The proof of this result very much relies on a study of the ring
structure of the set of clopen subspaces of $X_K$. This is done
in the first two parts of Section \ref{sec3}, with the last part
containing the proof of Theorem \ref{wittsap} and a slight generalization
of a result of Koprowski about the Witt equivalence of real algebraic function
fields over real closed fields, see Corollary \ref{koprocor}.

In the last two sections, we adapt the ideas used in Section \ref{sec2}
and \ref{sec3} to other situations. In the case of reduced Witt rings,
it turns out that analogs of Theorem \ref{wittiso} and \ref{wittsap} can be proved:
see Theorem \ref{wittred} and Corollary \ref{wittredsap}. Section \ref{sec4} is
dedicated to the proof of these two results after recalling some elementary facts
about preorderings and reduced Witt rings. 

In Section \ref{sec5}, we show how to generalize
Theorem \ref{wittiso} in the case of the Witt ring of hermitian forms
over a quadratic extension of a real field endowed
with its nontrivial automorphism, and the Witt group of hermitian forms over 
a quaternion division
algebra central over a real field and endowed with its canonical involution. 
The corresponding results are stated
as Theorem \ref{wittquad} and Theorem \ref{wittquat}. As for usual Witt rings, 
an important step in the proof of each of these results is to determine bijections
between certain subspaces of the space of orderings 
and certain subsets of the spectra of the Witt rings
adapted to each situation.

\section{Witt rings}\label{sec2}

\indent

Before turning to the proof of Theorem \ref{wittiso}, let us fix
some terminology and recall some basic facts about Witt rings and
orderings. For further informations, we refer to \cite{lam2}.

\subsection{Notations and preliminary results}

\indent

Let $K$ be a field. The notation $\l a_1,\cdots,a_n\r$ will refer to 
the diagonal quadratic form $a_1 X_1^2 + \cdots + a_n X_n^2$ where
the $a_i$'s belong to $K^*$. We shall denote by $W(K)$ the \emph{Witt ring}
of $K$ and by $I(K)$ its fundamental ideal. The
$n$th power of the fundamental ideal shall be denoted by $I^n(K)$.
It is additively generated by the $n$-\emph{fold Pfister forms}
$$ \pff{ a_1,\cdots,a_n}:= \l 1,-a_1 \r \otimes \cdots \otimes \l
1,-a_n \r.$$ The \emph{signed discriminant} of a (diagonal)
quadratic form $\l a_1,\cdots,a_n\r$ is defined to be the following
element of $K^*/{K^*}^2$:
$$(-1)^{\frac{n(n+1)}{2}}a_1 \cdots a_n.{K^*}^2.$$ It induces
a group isomorphism $d_{\pm} : I(K)/ I^2(K) \simeq K^*/{K^*}^2$.

An \emph{ordering} on $K$ is a subset $P \subsetneq K$ such that
$P+P \subseteq P$, $P.P \subseteq P$, $P \cup -P = K$. It follows that
$-1 \notin P$ and that $P\cap-P = \lbrace 0\rbrace$.  
>From the very definition, any ordering $P$ contains $\sigma(K)$. 
By Artin-Schreier theory, a field $K$ has an ordering if and only if
$-1\not\in\sigma(K)$, in which case $K$ is called {\em formally real}, or
{\em real} for short.  Furthermore, $\sigma(K) =\bigcap P$ where $P$ runs 
over all orderings of $K$. If $P$ is an ordering and $x\in K^*$, then
$x$ is said to be \emph{positive} (resp. \emph{negative}) at $P$, denoted
by $x \pos 0$ (resp. $x\nega 0$), if $x \in P$ (resp. $-x \in
P$). Note that
we can talk about the positivity or the negativity of any square
class, a square being always positive at any ordering. The set $X_K$ 
of all orderings on $K$ is a
topological space with a subbasis given by the clopen sets
$$H(a):= \lbrace P\in X_K \ | \ a \in P\rbrace,$$
where $a$ ranges over $K^*$. This topology is known as
the \emph{Harrison topology}.

The signature of a quadratic form $q$ at an ordering $P$ shall be
denoted by $\sgn_P (q)$. It induces a surjective ring
homomorphism $\sgn_P : W(K) \to \mathbb{Z}$.

Recall that by Pfister's local-global principle, the torsion
quadratic forms over a real field $K$ are exactly the forms $q$ 
with total signature zero,
i.e. $\sgn_P(q)=0$ for all $P\in X_K$.  Hence, $u(K)$, if finite, is
always even for a real field $K$.

A prime ideal $I$ in a ring $R$ is said to be of characteristic $p$
where $p$ is a prime number (resp. $0$) if the quotient ring $R/I$ has
characteristic $p$ (resp. $0$) and
we will say that ${\rm char} \ I=p$ (resp. $0$). In the case where
$R=W(K)$, we easily see that a prime ideal $I$ has characteristic
$p$ (resp. 0) if and only if the quotient $W(K)/I$ is isomorphic (as
a ring) to $\mathbb{Z}/p\mathbb{Z}$ (resp. $\mathbb{Z}$), see
\cite[Chapter VIII]{lam2}. Following \cite[Chapter VIII]{lam2}, let
$$ Y_K := \lbrace I \in \Spec\pr W(K) \prt \ | \ {\rm char} \ I = 0
\rbrace.$$ 
\begin{thm}[Harrison]\label{harrison}
There is a bijection between $X_K$ and $Y_K$.
\end{thm}
\begin{proof}
The bijection is defined as follows. To any $P \in X_K$, we can
associate the ideal $I_P:= \ker\sgn_P \in Y_K$. Conversely, from
$I\in Y_K$, one can construct an ordering by setting
$P_I:= \lbrace 0 \rbrace \cup \lbrace a \in K^* \ | \ \l a \r
\equiv 1 \mod I\rbrace$. See \cite[Chapter VIII, Proposition
7.4]{lam2} for more details.
\end{proof}

\subsection{Proof of Theorem \ref{wittiso}}

\indent

During the proof of Theorem \ref{wittiso}, we will need the following Lemma:

\begin{lem}\label{minus}
Let $K$ and $L$ be real fields. Suppose that there is a 
pair of maps $(t,T)$ with a group
isomorphism $t :{K}^*/{{K}^*}^2\to {L}^*/{{L}^*}^2$ and
a bijection $T: X_K \to X_L$ such that $x$ is
positive at $P$ if and only if $t(x)$ is positive at $T(P)$ for all
$x \in {K}^*/{{K}^*}^2$ and for all $P \in X_K$. Then there
exists a pair $(t,T)$ with the above properties and in addition
$t(-1)=-1$.
\end{lem}
\begin{proof}
By assumption, $-t(-1)$ is positive at all orderings 
of $L$, so $-t(-1) \in \dot{\sigma}(L)$, say $t(-1)= - \sum_{i=1}^n {u_i}^2$.
Now if $\sum_{i=1}^n {u_i}^2\neq 1\in L^*/L^{*2}$, then
let $\lbrace -1, - \sum_{i=1}^n {u_i}^2, \cdots \rbrace$ be
a $\mathbb{F}_2$-basis of the $\mathbb{F}_2$-vector space 
$L^*/{L^*}^2$. Let $\tilde{t} : L^*/{L^*}^2 \to L^*/{L^*}^2$ 
be the $\mathbb{F}_2$-vector space automorphism which exchanges
$-1$ and $- \sum_{i=1}^n {u_i}^2$ but fixes all other basis vectors. 
Let $t_1=\tilde{t}\circ t$.
Now, $t_1$ is a group isomorphism between $K^*/{K^*}^2$ and $L^*/{L^*}^2$ and
$t_1(-1)=-1$. Moreover, by decomposing any element $t(x) \in L^*/{L^*}^2$ 
in the previous $\mathbb{F}_2$-basis,
we easily see that $t(x)$ is positive at $Q$ if and only if $t_1(x)$ is positive at
$Q$ for any $Q\in X_L$.
\end{proof}

\noindent \textbf{Proof of Theorem \ref{wittiso}:} 
$(1) \Rightarrow (2)$ We will use the notations of the proof of
Theorem \ref{harrison}. Let $\Phi$ be a ring isomorphism between
$W(K)$ and $W(L)$. By Theorem \ref{harriso}, there exists a
group isomorphism $t: K^*/{K^*}^2 \to
L^*/{L^*}^2$ with $t(-1)=-1$. 
%%such that $\l 1,-x,-y,xy \r$ is hyperbolic 
%%over $K$ if and only if  $\l 1,-t(x), -t(y), t(x)t(y)\r$ is hyperbolic over
%%$L$, for all $x, \ y \in {K}^*$. 
More precisely, if $a\in K^*/{K^*}^2$, $t(a)$ is defined to be the signed
discriminant of $\Phi ( \l 1, -a \r)$: 
$t(a)=d_{\pm}(\Phi ( \l 1, -a \r))\in L^*/{L^*}^2$.

Now, we show how to construct $T$. As $\Phi$ is a ring isomorphism,
it is easy to see that $I \in Y_K$ if and only if $\Phi(I)\in Y_L$.
Now, consider the following diagram:
$$\xymatrix@R=40pt@C=40pt{X_K \ar@<2pt>[r]^{\normalsize\mbox{$\lambda_K$}}
\ar@{.>}[d]^{\normalsize\mbox{$T$}} \comm &
Y_K\ar@<2pt>[l]^{\normalsize\mbox{$\mu_K$}}\ar[d]^{\normalsize\mbox{$\Phi$}}\\
X_L\ar@<2pt>[r]^{\normalsize\mbox{$\lambda_L$}} &
Y_L\ar@<2pt>[l]^{\normalsize\mbox{$\mu_L$}}}$$ %@R=60pt@C=40pt
In this diagram, $\lambda_*:\, X_*\to Y_*:\,P\mapsto I_P$ is the 
bijection in Theorem
\ref{harrison}, and $\mu_*:\,Y_*\to X_*:\,I\mapsto P_I$ is its inverse. 

Define $T=\mu_L\circ\Phi\circ\lambda_K: X_K \to X_L$. More precisely, 
if $P \in X_K$, then 
$T(P):= \lbrace 0 \rbrace \cup \lbrace b\in L^* \ | \ \l b\r \equiv
1 \mod \Phi (I_P)\rbrace$. In particular, $q \in I_P$ if and only
if $\Phi(q) \in I_{T(P)}$.

Next, we show the compatibility of $(t,T)$ with respect to the
positivity at each ordering. Let $x \in {K}^*/{{K}^*}^2$ and $P \in
X_K$ such that $x\pos 0$. This means that $\l 1,-x \r\in I_P$, thus
$\Phi(\l 1,-x\r) \in I_{T(P)}$. By construction, $t(x)$ is the
signed discriminant of $\Phi(\l 1,-x\r)$. Let $\Phi(\l x \r) = \l
b_1, \cdots, b_{2n+1}\r$ (as a ring homomorphism, $\Phi$ sends the
units of $W(K)$ onto the units of $W(L)$ which are necessarily 
Witt classes of forms of odd dimension). We easily verify that
$$t(x)= (-1)^n b_1 \cdots b_{2n+1}.$$
As
$$0={\sgn}_{T(P)} \pr \Phi (\l 1,-x\r) \prt={\sgn}_{T(P)} ( \l 1,
-b_1, \cdots, -b_{2n+1}),$$ there are exactly $n$ (resp. $n+1$)
$b_i$'s which are negative (resp. positive) at $T(P)$. Therefore,
$t(x)$ is positive at $T(P)$. The converse is similar.

Lastly, we show that $T$ is in fact a homeomorphism. To show that
$T$ is continuous with respect to the corresponding Harrison
topologies, it suffices to show that $T^{-1}\pr H(b)  \prt$ is open in
$X_K$ for every $b \in L^*$. Let $b \in L^*$ and $a \in K^*$ be such
that $t(a)=b$. Now for each $P\in X_K$, we have that $a$ is positive at $P$ 
if and only if $b$ is positive at $T(P)$. This readily implies that
$T^{-1} \pr H(b)\prt  = H(a)$ thus proving that the bijection $T$ is 
continuous. Similarly,
$T^{-1}$ is continuous and $T$ is a homeomorphism.\\

\noindent $(2)\Leftrightarrow (3)$ If (3) holds, then clearly $T(H(a))=H(t(a))$ for
all $a\in K^*$, and by the definition of the Harrison topology, $T$ is clearly
a homeomorphism which establishes (2).  The converse is trivial.\\

Finally, assume that $u(K), u(L)\leq 2$.

\noindent $(3) \Rightarrow (1)$ The signature at an ordering of
any $n$-fold Pfister form is $0$ or $2^n$, so we have
\begin{eqnarray}\label{equ1}
{\sgn}_P\pr \pff{ x,y} \prt & = & {\sgn}_{T(P)}\pr \pff{
t(x),t(y)}\prt,
\end{eqnarray}
for all $x, \ y\in {K}^*/{{K}^*}^2$ and for all $P \in X_K$. If $x,
\ y\in {K}^*/{{K}^*}^2$, as $u(K)\leq 2$, the $2$-fold Pfister form
$\pff{ x,y}$ is hyperbolic if and only if $\pff{ x,y}$ is
torsion which in turn is equivalent to the fact that ${{\rm
sgn}}_{P}\pr \pff{ x,y}\prt=0$ for every $P\in X_K$, by Pfister's
local-global principle (see \cite[Chapter VIII, Theorem 3.2]{lam2}).
Using (\ref{equ1}), Pfister's local-global principle and the fact
that $u(L) \leq 2$, we obtain that the $2$-fold Pfister form $\pff{
x,y}$ is hyperbolic over $K$ if and only if the $2$-fold Pfister
form $\pff{ t(x),t(y)}$ is hyperbolic over $L$. Now, we get
condition $(1)$ by using Lemma \ref{minus} and Theorem \ref{harriso}.
\hfill$\Box$\bigskip
\begin{rems}\label{rema}
We have $u(K) \leq 2$ if and only if $I^2 (K)$ is torsionfree (see
\cite[Chapter XI, Proposition 6.26 (2)]{lam2}). For example, if $K$
is an extension of transcendence degree $1$ over a real closed field $k$,
then $u(K)\leq 2$
(see \cite[Theorems E, I]{elp}).
\end{rems}

\section{SAP fields}\label{sec3}

\indent

The purpose of this section is to prove isomorphism criteria for Witt rings
over SAP fields. 
We first study the space of orderings of such a field.

\subsection{Homeomorphisms of spaces of orderings of real fields}

\indent


Let $\mathcal{X}$ be a topological space that is boolean 
($=$ compact, totally disconnected and Hausdorff).
Then the set of clopen subsets
$\Clop(\mathcal{X})$ is a boolean ring with addition given by the 
symmetric difference $\symdif$ and multiplication given by the
intersection $\cap$.   It is clear that homeomorphic boolean spaces
$\mathcal{X}$ and $\mathcal{X}'$ give rise to isomorphic boolean
rings  $\Clop(\mathcal{X})$ and $\Clop(\mathcal{X}')$.

Furthermore, if $B$ is a boolean ring then
$\Spec(B)$ becomes a boolean topological space under the
Zariski topology, and again 
it is clear that isomorphic boolean rings $B$ and $B'$ give rise to
homeomorphic boolean spaces $\Spec(B)$ and $\Spec(B')$.

It is a consequence of Stone duality that any boolean space 
$\mathcal{X}$ is homeomorphic to  $\Spec(\Clop(\mathcal{X}))$
(see \cite{stone}).

Since spaces of orderings of fields are boolean spaces under the Harrison
topology, we can apply the above to get
\begin{cor}\label{homeo}
Let $K$ and $L$ be real fields. Then the following are equivalent:
\begin{enumerate}
\item[(1)] There is a homeomorphism $\Phi : X_K \to X_L$.
\item[(2)] There is a ring isomorphism $\Psi : (\Clop(X_K), \symdif, \cap) \to
(\Clop(X_L),\symdif, \cap)$.
\end{enumerate}
\end{cor}



\subsection{The group structure of $\pr K^* / \dot{\sigma}(K), \times\prt$ for SAP fields}

\indent

\begin{defi}\label{SAP-def}
A field $K$ satisfies the Strong Approximation Property (we also say
that $K$ is SAP for short) 
if one of the following equivalent conditions holds:
\begin{enumerate}
\item[(1)] For any two disjoint closed subsets $A, \ B$ of $X_K$, there exists an 
$a \in K^*$ such that $a$ is positive at every $P\in A$ (i.e. $A\subset H(a)$)  
and negative at every $P \in B$ (i.e. $B\subset H(-a)$).
\item[(2)] If $A$ is a clopen subset of $X_K$, then $A=H(a)$ for some $a \in K^*$.
\end{enumerate}
\end{defi}\noindent
Note that a nonreal field may be called SAP as both conditions are empty.
It is known that algebraic number fields, fields of transcendence degree 
at most $1$ over a real closed
field or the field of Laurent series in one variable over any 
uniquely ordered real field are SAP fields.

In the sequel, for any $a\in K^*$, we put
$$ H^-(a):= \{ P\in X_K \ | \ -a \in P \}\ .$$
Then $H^-(a) = H(-a) \in \Clop(X_K)$, and
we have $H^{-}(a)\symdif H^{-}(b)=H^{-}(ab)$.  

The purpose of this subsection is to give two descriptions of the 
multiplicative group $K^* / \dot{\sigma}(K)$
in the case where $K$ is an SAP field. 
For us, the more important one
is the following.
\begin{lem}\label{sum}
Let $K$ be real and SAP. Then the map
$$t :\ K^*/ \dot{\sigma}(K)\to  (\Clop(X_K), \symdif)\ :\ a\mapsto H^-(a)$$
is a well-defined group isomorphism.
\end{lem}
\begin{proof}
Clearly, $t$ is well-defined,  and it is surjective as $K$ is SAP. Moreover
$H^-(a) = H^-(b)$ if and only if $ab$ is positive at all $P\in X_K$ if and only if
$ab \in \dot{\sigma}(K)$ by Artin's Theorem. This proves that $t$ is bijective.
Also, $H^-(a) \symdif H^-(b) = H^-(ab)$, thus proving that $t$ is a 
group isomorphism.
\end{proof}

\begin{rem}
In general, one always has a group isomorphism 
$K^*/ \dot{\sigma}(K)\simeq  (\mathcal{H} ,\symdif)$ as above
(where $\mathcal{H}$ denotes the set of all $H^-(a)$) as was
first noticed by Craven in \cite{craven}. In the case where $K$ is SAP, 
we obviously have $\mathcal{H} = \Clop(X_K)$.
\end{rem}

For $n \in \mathbb{N}$, denote by $I^n_t (K)$ the ideal of torsion forms 
in $I^n(K)$, i.e. $I^n_t(K)=I^n(K)\cap W_t(K)$.
The following result
(which won't be used in the sequel) is another way to describe the group 
$K^* / \dot{\sigma}(K)$ under a more restictive hypothesis.
\begin{prop}\label{coho}
Let $K$ be a field with with $I^n_t (K) =0$ for some $n\in \NN$.
\begin{enumerate}
\item[(1)] The map 
$$\Phi : K^*/\dos(K)\to H^n (K, \mu_2)\,:\,a\bmod\dos(K)\to 
(a) \cup \underbrace{(-1) \cup \cdots \cup (-1)}_{n-1}$$
is well defined
and an injective group homomorphism.
\item[(2)] $\Phi$ is an isomorphism if and only if $K$ is SAP.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Let us for short write $(a_1,\cdots,a_n)$ for the symbol
$(a_1)\cup\cdots \cup (a_n)\in H^n (K, \mu_2)$.  Using the injectivity
of the Milnor map $e_n\,:\,I^nK/I^{n+1}K\to H^n (K, \mu_2)$ on $n$-fold
Pfister forms, we get that
\begin{eqnarray*}
\lefteqn{(a,-1,\cdots, -1)=(b,-1,\cdots, -1)}\\
 & \Longleftrightarrow
& \pff{a,-1,\cdots, -1}\cong \pff{b,-1,\cdots, -1}\\
 & \Longleftrightarrow &
\sgn_P(\pff{a,-1,\cdots, -1}-\pff{b,-1,\cdots, -1})=0\quad
(\forall P\in X_K)\\
 & \Longleftrightarrow & 
 ab\in\dos(K)
\end{eqnarray*}
(in the second equivalence, we use that 
$\pff{a,-1,\cdots, -1}-\pff{b,-1,\cdots, -1}\in I^nK$ together with
Pfister's local-global principle and the fact that $I^n_tK=0$).
This shows
well-definition and injectivity of $\Phi$.  It is clear that
$\Phi$ is then also a group homomorphism.

(2) Suppose in addition that $K$ is SAP.
Since $e_n$ is also surjective (by the Milnor conjecture), to show
surjectivity of $\Phi$, it
suffices to show that each $n$-fold $\pff{a_1,\cdots,a_n}$ is isometric
to $\pff{a,-1,\cdots, -1}$ for some $a\in K^*$. Since $I^n_tK=0$
and by Pfister's local-global principle, it suffices to show that
there exists $a\in K^*$ such that $H^-(a_1) \cap \cdots \cap H^- (a_n) = H^-(a)$,
but this clearly holds for SAP fields.

Conversely, suppose $K$ is not SAP.  Then there must exist
$a_1,a_2\in K^*$ such that for any $a\in K^*$, we have
$H^-(a_1) \cap H^-(a_2)\neq H^-(a)$.  In particular, for any $a\in K^*$
there exists $P\in X_K$ with
$\sgn_P(\pff{a_1,a_2,-1,\cdots, -1}-\pff{a,-1,\cdots, -1})\neq 0$.
By similar arguments as before, we conclude that
$(a_1,a_2,-1,\cdots, -1)$ is not in the image of $\Phi$.
\end{proof}





\subsection{Isomorphism criteria for the Witt ring of an SAP field}

\indent

The purpose of this subsection is to prove two isomorphism criteria
for the Witt rings of certain SAP fields. We first begin by the proof of Theorem
\ref{wittsap}.\\

\noindent \textbf{Proof of Theorem \ref{wittsap}:}
$(1) \Rightarrow (2)$ By Theorem \ref{wittiso}, 
there is a pair of maps $(t,T)$, where $t$ is a group
isomorphism $t :{K}^*/{{K}^*}^2\to {L}^*/{{L}^*}^2$ and $T$ is a homeomorphism
$T: X_K \to X_L$ such that $x$ is positive at $P$ if and only if $t(x)$ is positive at $T(P)$,
for all $x \in {K}^*/{{K}^*}^2$ and for all $P \in X_K$. By Artin's Theorem, it easily follows
that $x \in \dot{\sigma}(K)$ if and only if $t(x) \in \dot{\sigma}(L)$, hence $t$ induces
a group isomorphism $\dot{\sigma}(K)/{K^*}^2 \simeq \dot{\sigma}(L)/{L^*}^2$.

$(2) \Rightarrow (1)$ Suppose that $\Phi : X_K \simeq X_L$ is a homeomorphism
and that $\beta: \dot{\sigma}(K)/{K^*}^2 \simeq \dot{\sigma}(L)/{L^*}^2$ is a group isomorphism.
As $K$ and $L$ are SAP fields, $\Phi$ induces a ring isomorphism 
$\Psi : \Clop(X_K) \simeq \Clop(X_L)$
by Corollary \ref{homeo}. Combining this with Lemma \ref{sum},
we deduce that there is a group isomorphism $\gamma : K^*/\dot{\sigma}(K)
\simeq L^*/\dot{\sigma}(L)$. 
We have the following diagram with exact rows (where $\pi_K$ and $\pi_L$ are
the canonical surjections):
$$\xymatrix{0 \ar[r] & \dot{\sigma}(K)/{K^*}^2 \ar[r]\ar[d]^{\normalsize\mbox{$\beta$}}
& K^*/{K^*}^2\ar[r]^{\normalsize\mbox{$\pi_K$}}\ar@{.>}[d]^{\normalsize\mbox{$\alpha$}}
& K^*/\dot{\sigma}(K)\ar[r]\ar[d]^{\normalsize\mbox{$\gamma$}} & 0\\
0 \ar[r] & \dot{\sigma}(L)/{L^*}^2 \ar[r] & L^*/{L^*}^2\ar[r]^{\normalsize\mbox{$\pi_L$}}
& L^*/\dot{\sigma}(L)\ar[r] & 0}$$
By the five-lemma, as $\beta$ and $\gamma$ are group isomorphisms, there exists a group
isomorphism $\alpha :K^*/{K^*}^2 \simeq L^*/{L^*}^2$ such that the above diagram
commutes.

Finally, we prove that $(\alpha, \Phi)$ respects the positivity. Let $x \in K^*/{K^*}^2$
be such that $x \pos 0$ for a $P \in X_K$. Then $\pi_K (x) \pos 0$ so 
$P \notin H^- \pr \pi_K(x)\prt$. As $\Psi \pr H^- (\pi_K(x)) \prt = H^- \pr \gamma
(\pi_K(x)) \prt$ we have $\gamma \pr \pi_K(x) \prt \underset{\Phi(P)}{>} 0$
hence $\alpha (x) \underset{\Phi(P)}{>} 0$. The converse is similar. Now,
we apply Theorem \ref{wittiso} to $(\alpha, \Phi)$ to conclude.
\hfill$\Box$\bigskip

As a consequence, one obtains the following result essentially
due to Koprowski in \cite{koprowski}:
\begin{cor}\label{koprocor}
Let $K(t)$ be the rational function field in one variable over a real closed
field $K$, and let $L/K(t)$ be a finite extension.
Then
$$\begin{array}{rl}
W(L)\cong W(K(t)) & \mbox{if $L$ is real;}\\
W(L)\cong W(K(t)(\sqrt{-1}\,)) & \mbox{if $L$ is nonreal and $\sqrt{-1}\in L$;}\\
W(L)\cong W(K(t)(\sqrt{-(t^2+1)}\,)) & 
\mbox{if $L$ is nonreal and $\sqrt{-1}\notin L$;}
\end{array}$$
All these cases are mutually exclusive.
\end{cor}
\begin{proof}
It is well known that $u(L)=u(K(t))=u(K(t)(\sqrt{-1}\,))=2$.  Indeed, 
$K(t)(\sqrt{-1}\,)=K(\sqrt{-1}\,)(t)$
and hence $L(\sqrt{-1}\,)$ is $C_1$ since $K(\sqrt{-1}\,)$ is algebraically closed, 
so $u(L(\sqrt{-1}\,))\leq 2$
and thus $u(L)\leq 2$ (see, e.g., \cite[Ch.~6, Thm.~2.12]{pfis}).
Clearly, $L^{*2}\neq \dos(L)$, so $u(L)=2$.  Also, such fields $L$ are always
SAP (see, e.g., \cite[Theorem I]{elp}).

Now one readily observes that the irreducible polynomials $t^2+a\in K[t]$,
$a\in K^{*2}$, are $\FF_2$-independent in $\dos(K(t))/K(t)^{*2}$ (considered
as $\FF_2$-vector space).
Thus, we get for the cardinalities that $\card(K(t))=\card(K)=\card(K^{*2})$ 
since $K$ is clearly infinite and Euclidean.  It follows that
$\card(\dos(K(t))/K(t)^{*2})=\card(K)$.  Now if $L$ is a finite algebraic
extension of $K(t)$, then clearly $\card(L)=\card(K(t))=\card(K)$.
By Kummer theory, there exists a subset $S\subset K^{*2}$ (again with
cardinality $\card(S)=\card(K^{*2})=\card(K)$) such that
$\{t^2+a\,|\ a\in S\}$ still forms a
$\FF_2$-independent subset in the $\FF_2$-vector space
$\dos(L)/L^{*2}$, and we immediately get that
$\card(\dos(L)/L^{*2})=\card(\dos(K(t))/K(t)^{*2})$.

Suppose first that $L$ is nonreal.  The above together with the fact that
both $K(t)(\sqrt{-1}\,)$ and $K(t)(\sqrt{-(t^2+1)}\,)$ are also nonreal shows that 
$$\begin{array}{rcl}
L^*/L^{*2} & \cong & 
K(t)(\sqrt{-1}\,)^*/K(t)(\sqrt{-1}\,)^{*2}\\
 & \cong &  
K(t)(\sqrt{-(t^2+1)}\,)^*/K(t)(\sqrt{-(t^2+1)}\,)^{*2}\end{array}$$ 
(since for nonreal fields $F$ we have $F^*=\dos(F)$).
 
Since  $u(L)=u(K(t)(\sqrt{-1}\,))=u(K(t)(\sqrt{-(t^2+1)}\,))=2$, any $2$-fold
Pfister form over any of these fields will be hyperbolic. Thus, in order
to invoke Harrison's criterion (Theorem \ref{harriso}), it suffices to check 
whether or not there exists such an isomorphism between the square class
groups that sends $-1$ to $-1$.  

If $-1\in L^{*2}$, then any
isomorphism $L^*/L^{*2}\cong K(t)(\sqrt{-1}\,)^*/K(t)(\sqrt{-1}\,)^{*2}$ will
do as in both fields, $-1=1$ modulo squares.  Hence, by
Harrison's criterion, $W(L)\cong W(K(t)(\sqrt{-1}))$.

Suppose that $-1\notin L^{*2}$.  One readily checks that
$-1\notin K(t)(\sqrt{-(t^2+1)}\,)^{*2}$. Since the cardinalities of the 
square class groups are the same, it is now possible to choose an
isomorphism between them that sends $-1$ to $-1$. Hence, by
Harrison's criterion, $W(L)\cong W(K(t)(\sqrt{-(t^2+1)}\,))$.

If $L$ is real, then we have $u(L)=u(K(t))=2$ and thus
$\dos(K(t))/K(t)^{*2}\cong
\dos(L)/L^{*2}$ by the above.  We also have $X_{K(t)}\cong X_L$ (see
Proposition in the Appendix).
Hence, $W(L)\cong W(K(t))$ by Theorem~\ref{wittsap}.

The fact that all these cases are mutually exclusive follows readily
from the fact that for the level $s$, we have $s(K(t))=\infty$,
$s(K(t)(\sqrt{-1}))=1$ and $s(K(t)(\sqrt{-(t^2+1)}\,))=2$, and that 
for any field $F$, the characteristic of $W(F)$ is either $0$ (if
$s(F)=\infty$, i.e. $F$ is real), or $2s(F)$ (if $s(F)<\infty$, i.e.
$F$ is nonreal).
\end{proof}

\begin{rem}  Koprowski \cite[Prop.~2.2, Cor.~4.2]{koprowski} 
gives the above classification in the case where the field of constants
of $L$ is real closed (i.e. $K$ is the field of constants), which 
obviously corresponds to the cases where $L$ is real or $L$ is nonreal
and $\sqrt{-1}\notin L$.  His proof does not explicitly make use of the
fact that in the case where $L$ is real, $X_{K(t)}\cong X_L$.  Instead,
his methods are based on the use of real places and the notion of
quaternion-symbol equivalence.  In view of Theorem \ref{wittsap} 
which works for all SAP fields of $u$-invariant $\leq 2$, 
it seemed therefore natural to base our proof on the above homeomorphism
of the respective spaces of orderings, and it became desirable to
have a self-contained proof of this homeomorphism.  Such a proof can be
found in the Appendix.
\end{rem} 


\section{Reduced Witt rings}\label{sec4}

\indent

In this section, we adapt Theorem \ref {wittiso} and \ref{wittsap}
to the case of reduced Witt rings. Before stating and proving our results, 
we recall some notations and basic facts. For further details, see
\cite{lam1}.

\subsection{Notations and preliminary results}

\indent

Let $K$ be a real field. Recall that $T$ is a \emph{preordering}
over $K$ if $T\subsetneq K$, $K^2\subset T$, $T+T\subseteq T$ and
$T.T\subseteq T$. For a preordering $T$ over $K$, we thus have
$\sigma(K) \subseteq T$. Note that $\sigma(K)$ is the smallest preordering 
in $K$ which is consequently called the 
\emph{weak preordering} of $K$. We write $X_K
/ T$ for the non-empty subset (see \cite[Corollary 1.4]{lam1})
of $X_K$ consisting of all orderings $P$ containing $T$. 
Note that $X_K / T$ is a closed space of
$X_K$ and is a topological space with the induced topology. Note
also that $T^*= T \setminus \lbrace 0 \rbrace$ is a subgroup of $K^*$.

Fix a preordering $T$ over $K$. A \emph{$T$-form} $\phi$ over $K$ is an
expression $\l a_1,\cdots, a_n\r_T$ where $a_1,\cdots, a_n \in K^*$.
If $P\in X_K/ T$, the $P$-signature of $\phi$ is just
the sum of the signatures at $P$ of
the $a_i$'s. Note that we can talk about the positivity of an
element of $K^*/T^*$ at $P\in X_K/ T$, an element of $T$
being always positive at such an ordering. Two $T$-forms $\phi, \
\phi'$ are \emph{$T$-isometric} and we write $\phi\simeq_T \phi'$ if
they have the same dimension and the same signatures at every
element of $X_K/ T$. A $T$-form is $T$\emph{-hyperbolic} if
$\sgn_P \phi =0$ at every $P\in X_K/ T$. As in the
absolute theory, the set of the $T$-isometry classes with the
orthogonal sum and the tensor product is a semi-ring. By definition,
the reduced Witt ring $W_T(K)$ is the Grothendieck ring of the
$T$-isometry classes of $T$-forms modulo the ideal generated by
$T$-hyperbolic forms.

By viewing a quadratic form $\l a_1,\cdots,a_n\r$ as a $T$-form, we
obtain a surjective ring homomorphism $W(K) \to W_T(K)$. The image
of $I(K)$ by this homomorphism is $I_T(K)$ which is nothing but the
ideal of $T$-isometry classes of even-dimensional $T$-forms. The
$n$th power of this ideal is denoted by $I^n_T (K)$ 
and is generated by the $T$-Pfister forms $\pff{
a_1,\cdots,a_n}_T$ (with the obvious notation). The signed
discriminant $d_{T,\pm}$ of a $T$-form $\phi=\l a_1,\cdots,a_n\r_T$ is
the following element of $K^*/T^*$:
$$(-1)^{\frac{n(n+1)}{2}} a_1\cdots a_n.T^*.$$
Moreover, it induces a group isomorphism $
d_{T,\pm}:I_T(K)/I_T^2(K) \simeq K^*/T^*$.

We say that two $T$-forms are chain $T$-equivalent if we can change
one to another by a finite sequence of transformations of the three
following types:
$$\begin{array}{lc}
(A) & \l a_1,\cdots,a_n\r_T \to \l t_1 a_1,\cdots ,t_n a_n\r_T, \quad
t_1,\cdots ,t_n \in T^*,\\[1ex]
(B) & \l a_1,\cdots,a_i,\cdots,a_j,\cdots,a_n\r_T
\to\\
 &  \l a_1,\cdots,a_i+a_j,
\cdots,a_i a_j(a_i+a_j),\cdots,a_n\r_T\quad
\mbox{}\hfill\mbox{(where $a_i+a_j\neq 0$).}\\[1ex]
(C) & \l a_1,\cdots,a_i,\cdots,a_j,\cdots,a_n\r_T
\to \l a_1,\cdots,a_j,\cdots,a_i,\cdots,a_n\r_T
\end{array}$$
If two $T$-forms are chain
$T$-equivalent, it is clear that they are $T$-isometric but the
converse also holds by Witt's Chain Equivalence Theorem:
\begin{thm}[{\cite[Theorem 1.28]{lam1}}]\label{eqch}
If two $T$-forms are $T$-isometric then they are chain $T$-equivalent.
\end{thm}

We denote by $Y_K^T$ the set of prime ideals of characteristic $0$
in $W_T(K)$. The following is an analogue of Theorem \ref{harrison}.
\begin{prop}\label{harred}
There is a bijection between $X_K / T$ and $Y_K^T$.
\end{prop}
\begin{proof}
The definition of the bijection is the same as in the proof of
Theorem \ref{harrison}. If $P \in X_K / T$, then
$I_P:= \ker (\sgn_P)$ is an element of $Y_K^T$. Conversely, if $I
\in Y_K^T$, then $P_I:=\lbrace 0\rbrace \cup \lbrace a\in K^* \ | \ \l
a\r_T \equiv 1 \mod I\rbrace$ is an element of $X_K / T$, and these two
maps are inverse to each other.
\end{proof}

\subsection{Statement and proof of the criterion}

\begin{thm}\label{wittred}
Let $T$ be a preordering over the real field $K$ and $S$ be a
preordering over the real field $L$. Then the following are
equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W_T(K) \simeq W_S(L)$.
\item[(2)] There is a group isomorphism $t: K^*/T^* \to
L^*/S^*$ sending $-1$ to $-1$ such that the
$T$-form $\l 1,-x,-y,xy \r_T$ is $T$-hyperbolic over $K$ if and only if the
$S$-form $\l 1,-t(x), -t(y), t(x)t(y)\r_S$ is $S$-hyperbolic over
$L$ for all $x, y \in K^*$.
\item[(3)] There is a pair of maps $(t,D)$ with a group
isomorphism $t :{K}^*/T^*\to L^*/S^*$ and a homeomorphism
$D: X_K / T \to X_L / S$ such that
$x$ is positive at $P$ if and only if $t(x)$ is positive at $D(P)$,
for all $x \in K^*/T^*$ and for all $P \in
X_K/ T$.
\item[(4)] There is a pair of maps $(t,D)$ with a group
isomorphism $t :{K}^*/T^*\to L^*/S^*$ and a bijection $D: X_K
/ T \to X_L / S$ such that
$x$ is positive at $P$ if and only if $t(x)$ is positive at $D(P)$,
for all $x \in K^*/T^*$ and for all $P \in
X_K/ T$.
\end{enumerate}
\end{thm}
\begin{proof} 
The equivalence of (1) and (2) is the ``reduced'' version of 
Harrison's Criterion.  It can be proved by readily adapting the
proofs of Harrison's criterion found, for example, in 
\cite[Ch. XII, Theorem 1.8]{lam2} or \cite{pscl} to the reduced setting,
using chain T-equivalence (Theorem \ref{eqch}) instead of the usual
chain equivalence, $d_{T,\pm}$ instead of $d_{\pm}$, etc.  
We leave the details to the reader.\\

$(1)\Rightarrow (3)$ Let $\Phi$ be a ring isomorphism
between $W_T(K)$ and $W_S(L)$. The signed discriminant induces  a
group isomorphism $t: K^*/T^* \simeq L^*/S^*$ defined by
$t(a):=d_{\pm}\pr \Phi (\l 1,-a\r_T)\prt$ for $a \in K^*/T^*$. By
Proposition \ref{harred}, we get the desired bijection $D$ between
$X_K/ T$ and $X_L / S$. The implication can now be shown by applying
the ``reduced'' analogs of the arguments used in the proof of $(1)
\Rightarrow (2)$ in Theorem \ref{wittiso}.\\

$(3)\Rightarrow (4)$ is clear.\\

$(4) \Rightarrow (2)$ First note that $(4)$ implies that
$t(-1)=-1$. Indeed, $-t(-1)\pos 0$ for each $P \in X_L/ S$
thus $$-t(-1) \in \bigcap_{Q\in X_L/ S} Q\quad =\ S,$$ 
(see \cite[Theorem 1.6]{lam1}).

Note also that the compatibility between $(t,D)$ and the positivity
readily implies that the $2$-fold $T$-Pfister form $\pff{ a,b }_T$ is
$T$-hyperbolic if and only if the $2$-fold $S$-Pfister form 
$\pff{t(a),t(b)}_S$ is $S$-hyperbolic.
\end{proof}
\begin{rems}
$(1)$ If $T$ and $S$ are the corresponding weak preorderings and if
$K, \ L$ are pythagorean, Theorem \ref{harred} is nothing but
Theorem \ref{harriso} in this particular case.\\
$(2)$ If $T$ and $S$ are the corresponding weak preorderings, then
we have the following ring isomorphism $W_T(K) \simeq W(K)/ W_t(K)$,
$W_S(L) \simeq W(L)/ W_t(L)$ (see \cite[Chapter 1]{lam1}) where
$W_t$ denotes the torsion subgroup of the corresponding Witt ring.
Hence Theorem \ref{harred} can be viewed as Harrison's criterion for
$W/W_t$ in this case.
\end{rems}

If $K$ and $L$ are two fields such that $W(K)\simeq W(L)$ as rings,
then, by the preceding remark, $W_T(K) \simeq W_S(L)$ as rings 
if $T$ (resp. $S$) is the
weak preordering of $K$ (resp. of $L$). The following
example shows that the converse is false:
\begin{ex}
Let $K$ and $L$ be any uniquely ordered fields with weak preorderings 
 $T$ and $S$, respectively.  Since every element of
$K$ (resp. $L$) is a sum of squares  or the negative of a 
sum of squares, one clearly has 
$W_T(K) \simeq W_S(L) \simeq \mathbb{Z}$.
But $W(K)$ and
$W(L)$ need not be isomorphic.  For example,
$\RR$ and $\QQ$ are uniquely ordered but obviously
$W(\RR)\not\simeq W(\QQ)$.
\end{ex}

\subsection{An isomorphism criterion for the reduced Witt ring of an SAP field}

\indent

Let $K$ be a real field and $T$ be a preordering on $K$. 
If we denote by $C(X_K / T, \mathbb{Z})$ the ring of continuous
functions from $X_K$ to $\mathbb{Z}$ (we endow $\mathbb{Z}$ with the discrete
topology), then the signatures induce a ring monomorphism
$$\sgn : W_T (F) \hookrightarrow C(X_K / T, \mathbb{Z}): 
q \mapsto \pr P \mapsto \sgn_P (q)\prt,$$
see \cite[\S 1]{lam1}. In fact, we can be more precise. 
Denote by ${\bf 1}$ the element
of $C(X_K / T, \mathbb{Z})$ which sends every ordering to $1$. 
The fact that $\sgn_P (q) \equiv \dim q \mod 2$ for every $P \in X_K$ and 
every $q \in W(K)$, 
implies that the previous map induces a ring monomorphism
$$\Phi_T : W_T (K) \hookrightarrow \mathbb{Z}.{\bf 1} + C(X_K / T, 2\mathbb{Z}).$$

In the case of SAP fields, we have the following 
criterion for reduced Witt rings:
\begin{cor}\label{wittredsap}
Let $K$ (resp $L$) be a SAP field and $T$ (resp. $S$) be a 
preordering over $K$ (resp. $L$). Then the following are equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W_T (K) \simeq W_S(L)$.
\item[(2)] There is a homeomorphism $X_K / T \simeq X_L / S$.
\item[(3)] There is a ring isomorphism 
$\mathbb{Z}.{\bf 1} + C(X_K / T, 2\mathbb{Z})
\simeq \mathbb{Z}.{\bf 1} + C(X_L / S, 2\mathbb{Z})$.
\end{enumerate}
\end{cor}
\begin{proof}
$(1) \Rightarrow (2)$ This is obvious by Theorem \ref{wittred}.\\
$(2) \Rightarrow (1)$ Let $D: X_K / T \simeq X_L / S$ be a homeomorphism. 
Then it induces a ring isomorphism 
$\widetilde{D} : \Clop (X_K / T) \simeq \Clop (X_L / S)$.
Define
$$\theta_K: \left\{
\begin{array}{cll}
K^*/T^* & \rightarrow & \Clop (X_K/ T)\\
a & \mapsto & H^{-}_T(a):= \lbrace P \in X_K / T \ | \ a\notin P\rbrace
\end{array}.\right.$$
It is easy to see that $\theta_K$ is well-defined, 
surjective as $K$ is a SAP field and
injective as $H^{-}_T(a) = H^{-}_T(b)$ if and only if $ab \in T^*$. 
Now we define the group isomorphism
$t:= {\theta_L}^{-1} \circ \widetilde{D} \circ \theta_K : K^*/T^* \simeq L^*/S^*$ 
(where
$\theta_L$ is defined in an analogous way as $\theta_K$).
Finally, let $x \in K^*/T^*$. We have 
$t(x) = {\theta_L}^{-1} \pr \widetilde{D}(H^{-}_T(x)) \prt= y$
where $\widetilde{D}(H^{-}_T(x)) = H^{-}_S(y)$. Let $P \in X_K / T$.
Then
\begin{eqnarray*}
x \ \textrm{is positive at} \ P & \iff & P \notin H^{-}_T(x)\\
                                & \iff & D(P) \notin H^{-}_S(y)\\
                                & \iff & y=t(x) \ \textrm{is positive at} \ D(P).
\end{eqnarray*}
Now we apply Theorem \ref{wittred} to conclude.

$(1) \iff (3)$ This is an immediate consequence of the fact that 
for any SAP field $K$ and any preordering $T$ on 
$K$, we have that the above map $\Phi_T: W_T(K) \to 
\mathbb{Z}.{\bf 1} + C(X_K / T, 2\mathbb{Z})$ is an isomorphism
(see \cite[Theorem 16.2]{lam1}).
\end{proof}

\section{Isomorphism criteria for Witt groups of hermitian forms}\label{sec5}

\indent

The ideas of the proof of Theorem \ref{wittiso} can be adapted
to two particular situations involving Witt groups of hermitian forms. 
First, we are interested in finding such a criterion for the Witt ring 
of hermitian forms over a quadratic field
extension with its nontrivial automorphism. 
In the second situation, we deal with
the Witt group of hermitian forms over a quaternion division 
algebra endowed with its canonical involution.
In each of these two cases, we have to recall some notations 
and notions and we refer
to \cite[Chapter VII, Chapter X]{scharlau} for further details.

\subsection{Witt rings of hermitian forms over quadratic field extensions}

\indent

Let $k$ be a field, $K/k$ be a quadratic field extension and
$\sigma_K$ be the nontrivial $k$-automorphism of $K$. Every
nondegenerate hermitian form over $(K,\sigma_K)$ can be
diagonalized and we will use the usual notation 
$\l a_1,\cdots,a_n\r$, $a_i\in k$, for such a diagonalization.
We denote by
$W(K,\sigma_K)$ the Witt ring of nondegenerate hermitian forms over
$(K,\sigma_K)$ and by $I(K,\sigma_K)$ its fundamental ideal. The
\emph{signed discriminant} of a hermitian form $\l a_1,\cdots,a_n\r$
is defined to be the following element of $K^*/N_{K/k}(K^*)$ (where
$N_{K/k}(K^*)$ denotes the usual norm group of $K/k$):
$$(-1)^{\frac{n(n+1)}{2}}a_1 \cdots a_nN_{K/k}(K^*).$$ 
It induces
a group isomorphism $d_{\pm} : I(K,\sigma_K)/ \pr
I(K,\sigma_K)\prt^2 \simeq K^*/N_{K/k}(K^*)$.

In \cite[Theorem 1.3]{gb}, the first author has proved the following
analogue of Theorem \ref{harriso} for Witt rings of hermitian forms over
quadratic field extensions:
\begin{thm}[Grenier-Boley]\label{main6}
Let $k$ and $l$ be two fields. Let $K=k(\sqrt{a})$ \emph{(}resp.
$L=l(\sqrt{b})$\emph{)} be a quadratic field extension of $k$
\emph{(}resp. $l$\emph{)}. Then, the following are equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W(K,\sigma_K)\simeq W(L,\sigma_L)$.
\item[(2)] There is a group isomorphism $t: {k}^*/N_{K/k}({K}^*) \to
l^*/N_{L/l}({L}^*)$ sending $-1$ to $-1$ such that the quadratic
form $\pff{ a,x,y }$ is hyperbolic over $k$ if and only if the
quadratic form $\l \l b,t(x),t(y)\r \r$ is hyperbolic over $l$ for
all $x, \ y \in {k}^*$.
\end{enumerate}
\end{thm}

Suppose now that $k$ is a real field and that $K=k(\sqrt{a})$ is a
quadratic field extension of $k$. Let
$$ X_k^a := \lbrace P\in X_k \ | \ a \nega 0 \rbrace = H(-a).$$
Note that, if $P \in X_k^a$ and if $x\in N_{K/k}(K^*)$ then $x$ is
positive at $P$. Consequently, we can talk about the positivity or
the negativity of any norm class at an ordering in $X_k^a$.

To each hermitian form $(V,h)$ over $(K,\sigma_K)$ one can associate
a $k$-quadratic form defined by $q_h(x):=h(x,x)$ for $x\in V$, where
$V$ is considered as a $k$-vector space.  Recall that if, over $K$, $h\simeq
\qf{a_1,\ldots ,a_n}$ (where $a_i\in k$), then, over $k$,
$q_h\simeq\pff{a}\otimes\qf{a_1,\ldots ,a_n}$.  The signature
of the hermitian form $h$ at $P\in X_k^a$ is then defined by
$\sgn_P(h):=\frac{1}{2}\sgn_P(q_h)$ which is an integer.
For each $P\in  X_k^a$, we thus get a surjective ring
homomorphism $\sgn_P:W(K,\sigma_K)\to\ZZ$.


We will also need the following notation:
$$ Z_k^a:= \lbrace I \in \Spec \pr W(K,\sigma_K)\prt\ |\ {\rm
char}\ I=0\rbrace.$$

As a first step toward Theorem \ref{wittquad}, we can prove the
following analogue of Theorem \ref {harrison} and Proposition \ref{harred}:
\begin{thm}\label{harquad}
There is a bijection between $X_k^a$ and $Z_k^a$.
\end{thm}
\begin{proof}
The bijection is defined in the same way as in the proof of Theorem
\ref{harrison}. We will sketch some of the details in this hermitian setting
for the reader's convenience.

First of all, a prime ideal $I$ of $W(K,\sigma_K)$ has
characteristic $0$ if and only if $W(K,\sigma_K)/I$ is isomorphic
(as a ring) to $\mathbb{Z}$. If $P \in X_k^a$, the ring homomorphism
$\sgn_P : W(K,\sigma_K) \to \mathbb{Z}$ is onto, hence
$I_P:=\ker(\sgn_P)$ is an element of $Z_k^a$.

If $I \in Z_k^a$, we put
$$P_I:= \lbrace 0 \rbrace \cup \lbrace c \in k^* 
\ | \ \l c\r\equiv 1 \mod I\rbrace,$$
(here, the notation $\l c \r$ refers to a one-dimensional hermitian
form). This is in fact an element of $X_k^a$. If $c\in k^*$ such
that $\l c\r\not\equiv 1 \mod I$ then $\l c \r\equiv
-1\mod I$ and hence $P_I \cup -P_I=k$. We clearly have $P_I.P_I
\subseteq P_I$ and $P_I \subsetneq k$. 
Let $c, \ d\in P_I$. Then the hermitian forms $\l c,
d\r$ and $\l c+d, (c+d)cd\r$ are isometric. Now $e:= c+d\neq 0$ and
$2 \equiv 2 \l e\r \mod I$. As $\chr I\neq 2$, this implies
that $e\in P_I$. Now the hermitian form $\qf{1,a}$ is hyperbolic
since the quadratic form 
$\pff{a}\otimes\qf{1,a}\simeq\qf{1,-a}\otimes\qf{1,a}$ is clearly
hyperbolic.  This implies that $a$ is negative at $P_I$, i.e.
$P_I\in X_k^a$.

Finally, one readily checks that the map $P\mapsto I_P$ gives the desired
bijection whose inverse is $I\mapsto P_I$.
\end{proof}
\begin{rem}
As a consequence of the previous result, 
one can determine the spectrum of the ring
$W(K,\sigma_K)$. It consists of the fundamental ideal 
of $W(K,\sigma_K)$ (which is the unique maximal ideal of 
characteristic $2$ in $W(K,\sigma_K)$), $\ker(\sgn_P)$ where $P\in X_k^a$
(which are exactly the prime ideals of characteristic $0$ in $W(K,\sigma_K)$) 
and $\ker(\sgn_P) \mod \ p$ where $p$ is a prime
number and $P\in X_k^a$. The proof of this fact can easily be adapted from
the determination of the spectrum of the usual Witt ring, 
see, e.g., \cite[Ch.VIII, Theorem 7.5]{lam2}.
\end{rem}

The analogue of Theorem \ref{wittiso} in the hermitian context 
is the following:
\begin{thm}\label{wittquad}
We keep the notations of Theorem \ref{main6} and we suppose further that
$k$ and $l$ are two real fields such that $u(k), \ u(l) \leq 6$.
The following are equivalent:
\begin{enumerate}
\item[(1)] There is a ring isomorphism $W(K,\sigma_K) \simeq
W(L,\sigma_L)$.
\item[(2)] There is a pair of maps $(t,T)$ with a group
isomorphism $t :{k}^*/N_{K/k}(K^*)\\ \to {l}^*/N_{L/l}(L^*)$ such
that $t(-1)=-1$, and a homeomorphism $T: X_k^a \to X_l^b$
such that, for all $x \in {k}^*/N_{K/k}(K^*)$ and for all $P \in X_k^a$, 
$x$ is positive at $P$ if and only if $t(x)$ is positive at
$T(P)$.
\item[(3)] There is a pair of maps $(t,T)$ with a group
isomorphism $t :{k}^*/N_{K/k}(K^*)\\ \to {l}^*/N_{L/l}(L^*)$ such
that $t(-1)=-1$, and a bijection $T: X_k^a \to X_l^b$ such
that, for all $x \in {k}^*/N_{K/k}(K^*)$ and for all $P \in
X_k^a$, $x$ is positive at $P$ if and only if $t(x)$ is positive at
$T(P)$.
\end{enumerate}
\end{thm}
\begin{proof} $(1)\Rightarrow (2)$
Let $\Phi$ be a ring isomorphism between $W(K,\sigma_K)$ and
$W(L,\sigma_L)$. By Theorem \ref{main6}, the map $t$ of condition
$(2)$ exists and satisfies $t(-1)=-1$ (more precisely, if $a\in
k^*/N_{K/k}(K^*)$, $t(a)$ is defined to be the (hermitian) signed
discriminant of $\Phi ( \l 1, -a \r)$).

As in the proof of Theorem \ref{wittiso}, using Theorem
\ref{harquad}, we construct a bijection $T$ between $X_k^a$ and
$X_l^b$. We also get the compatibility of $(t,T)$ with respect to
the positivity at each ordering in the same manner.

Lastly, $X_k^a$ is a topological space with its topology induced by
Harrison's topology over $X_k$. An open subbasis of $X_k^a$ is given
by
$$H^a(c):= \lbrace P\in X_k^a \ | \ c \in P\rbrace = H(c)\cap H(-a),$$
where $c$ ranges over $k^*$. $T$ is then readily seen to be a
homeomorphism for the induced topologies.\\

$(2)\Rightarrow (3)$ This is clear.\\

$(3) \Rightarrow (1)$ Condition $(3)$ and the fact that $u(k), \
u(l) \leq 6$ implies that the quadratic form $\pff{a,x,y}$ is
hyperbolic over $k$ if and only if the quadratic form 
$\pff{b,t(x),t(y)}$ is hyperbolic over $l$. Now we apply Theorem
\ref{main6} to deduce condition $(1)$.
\end{proof}

\begin{rem}
Fields whose $u$-invariant is at most $6$ include global fields and fields
of transcendence degree at most $2$ over a real closed field
(see, e.g., \cite[Ch.8, Theorem 2.12]{pfis}).
\end{rem}

\subsection{The Witt group of a quaternion division 
algebra with its canonical involution}

\indent

Let $Q$ be a quaternion division algebra over a field $K$ 
with canonical involution $\gamma$.
We denote by $W(Q,\gamma)$ the Witt group of 
nondegenerate hermitian forms over $(Q,\gamma)$.
It is easy to see that $W(Q,\gamma)$ is in fact a $W(K)$-module. 
The goal of this Subsection
is to find an isomorphism criterion for Witt modules of 
quaternion division algebras
over the same real base field $K$. For this purpose, 
we have to find the object
corresponding to the spectrum of the Witt ring
in the case of this particular Witt module. 
This object is in fact nothing but a subset of the support
of this module.

Let $A$ be a commutative ring and let $M$ be an $A$-module. 
If $I \in \Spec (A)$, we denote by
$M_I$ the localized module of $M$ at $I$. 
The support of $M$ is a subset of $\Spec(A)$ defined as follows:
$$ \Supp(M) = \lbrace I \in \Spec(A) \ | \ M_I \neq 0 \rbrace.$$
Suppose now that $A=W(K)$, and that $M=W(Q,\gamma)$ 
as above. Then $M$ is an $A$-module and we set:
$$ \Supp^0 (M) = \lbrace I \in \Supp (M)\ | \chr I =0\rbrace.$$
When $K$ is a real field and $a, \ b \in K^*$, we define
$$X_K^{a,b} = \lbrace P \in X_K \ | \ a \nega 0 \textrm{ and } b \nega 0\rbrace
= H(-a)\cap H(-b).$$
We have the following description which is similar to the description
obtained in Theorem \ref{harrison} and \ref{harquad}:
\begin{prop}\label{harquat}
Let $Q=(a,b)_K$ be a quaternion division algebra 
endowed with its canonical involution
$\gamma$ over a real field $K$ and $M$ be the Witt group of $(Q,\gamma)$.
Then there is a bijection between $\Supp^0 (M)$ and $X_K^{a,b}$.
\end{prop}
\begin{proof}  
Recall that any nondegenerate hermitian form $h$ over $(Q,\gamma)$ can be
diagonalized as $h\simeq\qf{a_1,\ldots, a_n}$ with $a_i\in K^*$.  Similarly
as in the case of hermitian forms over quadratic extensions, we can define
the signature of $h$ at $P$ for every $P\in X_K^{a,b}$ using the fact that
for any $P\in X_K^{a,b}$ and any $x\in \Nrd_{Q/K}(Q^*)$  we have $x\in P$.
For each $P\in X_K^{a,b}$, we thus get a surjective ring homomorphism 
$\sgn_P:W(Q,\gamma)\to\ZZ$.

To each $P \in X_K^{a,b}$ we associate $I_P=\ker(\sgn_P)$. We have
to show that $I_P \in \Supp^0 (M)$, that is $M_{I_P} \neq 0$.
Now $M_{I_P} \neq 0$ if and only if there exists $h \in M$ such that for every
$s \in W(K) \setminus I_P$, $s.h \neq 0$. We choose $h=\qf{1}$ 
and we let $s \in W(K) \setminus I_P$. If $s.h$ is hyperbolic 
as a hermitian form over $(Q,\gamma)$ then
the quadratic form $\pff{a,b} \otimes s$ is hyperbolic 
as a quadratic form over $K$.
Now the hypotheses imply that $\sgn_P \pr \pff{ a,b} \otimes s\prt \neq 0$ which
is a contradiction. This means that $I_P \in \Supp^0 (M)$.

Conversely, to each $I \in \Supp^0 (M)$ we associate
$$P_I:=\lbrace 0 \rbrace \cup \lbrace x \in K^* \ | \ \l x \r
\equiv 1 \mod I\rbrace\ .$$ 
We have to show that $P_I \in X_K^{a,b}$.
Suppose otherwise, i.e. $P_I \notin X_K^{a,b}$. 
Without loss of generality, we may assume
that $a$ is positive at $P_I$. This means that the quadratic form 
$\l 1,a \r$ is in $W(K) \setminus I$.
Let $h \in M$ and $s= \l 1,a \r$. We are going to show that $s.h=0$. 
We may assume that
$\rank(h) =1$ and therefore it suffices to show that the quadratic form 
$\pff{a,b} \otimes s\simeq\qf{1,-a}\otimes\qf{1,-b}\otimes\qf{1,a}$ 
is hyperbolic which is obvious. This implies that $M_I=0$ 
which is a contradiction. Thus,
$P_I \in X_K^{a,b}$.

Finally, one readily verifies that the above defined maps $P\mapsto I_P$ and
$I\mapsto P_I$ yield bijections between $X_K^{a,b}$ and $\Supp^0 (M)$
that are inverse to each other.
\end{proof}
\begin{rem}
With the above notations, the support of $W(Q,\gamma)$ consists of
$I(K)$, $\ker(\sgn_P)$ where $P \in X_K^{a,b}$, $\ker(\sgn_P) \mod p$
where $p$ is a prime number and $P \in X_K^{a,b}$. Again, the proof is similar
to the determination of the spectrum of $W(K)$ and is omitted.
\end{rem}

For quaternion division algebras over the same base field, 
the first author has proved
the following criterion in \cite[Corollary 1.4]{gb}:
\begin{prop}[Grenier-Boley]\label{main8}
Let $Q_1=(a,b)_K$ (resp. $Q_2=(c,d)_K$) be a 
quaternion division algebra over $K$ endowed
with its canonical involution $\gamma_1$ 
(resp. $\gamma_2$). Then, the following are equivalent:
\begin{enumerate}
\item[(1)] There is a $W(K)$-module isomorphism 
$W(Q_1,\gamma_1)\simeq W(Q_2,\gamma_2)$.
\item[(2)] There is a group isomorphism $t :{K}^*/ \Nrd_{Q_1/K}({Q_1}^*)
\simeq {K}^*/ \Nrd_{Q_2/K}({Q_2}^*)$
with $t(-1)=-1$ such that the quadratic form $\pff{a,b,u,v}$ 
is hyperbolic over $K$ if and only if the quadratic
form $\pff{c,d,t(u),t(v)}$ is hyperbolic over
$K$ for all $u, \ v \in {K}^*$, where $\Nrd_{Q_i/K}({Q_i}^*)$
denotes the group of reduced norms from the quaternion algebra 
$Q_i$ for $i=1, \ 2$.
\end{enumerate}
\end{prop}

We are now able to state:

\begin{thm}\label{wittquat}
Let $K$ be a real field with $u(K) \leq 14$. Let $Q_1=(a,b)_K$, 
$Q_2=(c,d)_K$ be quaternion division algebras over $K$
endowed with their respective canonical involutions 
$\gamma_1$ and $\gamma_2$. Then
the following are equivalent:
\begin{enumerate}
\item[(1)] There is a $W(K)$-module isomorphism 
$W(Q_1,\gamma_1) \simeq W(Q_2,\gamma_2)$.
\item[(2)] There is a pair of maps $(t,T)$ with a group isomorphism 
$t : K^*/ \Nrd_{Q_1}(Q_1^*)\\ \to K^*/ \Nrd_{Q_2}(Q_2^*)$ such
that $t(-1)=-1$, and a homeomorphism $T: X_K^{a,b} \to X_K^{c,d}$
such that, for all $x \in K^*/\Nrd_{Q_1}(Q_1^*)$ and for all 
$P \in X_K^{a,b}$, $x$ is positive at $P$ if and only if 
$t(x)$ is positive at $T(P)$.
\item[(3)] There is a pair of maps $(t,T)$ with a group isomorphism 
$t : K^*/ \Nrd_{Q_1}(Q_1^*)\\ \to K^*/ \Nrd_{Q_2}(Q_2^*)$ such
that $t(-1)=-1$, and a bijection $T: X_K^{a,b} \to X_K^{c,d}$
such that, for all $x \in K^*/\Nrd_{Q_1}(Q_1^*)$ and for all 
$P \in X_K^{a,b}$, $x$ is positive at $P$ if and only if 
$t(x)$ is positive at $T(P)$.
\end{enumerate}
\end{thm}
\begin{proof}
The proof is similar to the proofs of Theorem \ref{wittiso} and \ref{wittquad}
by using Proposition \ref{harquat} and Proposition \ref{main8}.  We
leave the details to the reader.
\end{proof}\vspace*{0.3cm}

\begin{center}{\Large \bf Appendix:}
\end{center}
\begin{center}{\Large \bf The space of orderings of a one-dimensional function field over~$R$}
\end{center}
\begin{center}
{\large \textsc{Claus Scheiderer}}
\end{center}

Let $R$ be a real closed field. We are going to show here that the
space of orderings of a real one-dimensional function field $K$ over
$R$ depends (up to homeomorphism) only on $R$, but not on $K$. This
is an easy application of general and well-known techniques from real
algebra.

As in the preceding text we denote the topological space of all
orderings of a field $K$ by $X_K$. Let $R(t)$ be the field of
rational functions in one variable over~$R$. Let $S\subset R$ be a
\sa\ set, i.e., a finite union of open intervals and singletons. By
$\wh S$ we denote the set of all orderings $P$ of $R(t)$ for which
there exist $a$, $b\in R\cup\{\pm\infty\}$ with $a<_Pt<_Pb$ and
$\left]a,b\right[\subset S$. The sets $\wh S$, for $S$ running
through the \sa\ subsets of $R$, are precisely the open-closed
subsets of the boolean topological space $X_{R(t)}$. The operator
$S\mapsto\wh S$ commutes with the finite boolean set-theoretic
operations, and $\wh S_1=\wh S_2$ holds if and only if the set-theoretic
difference of $S_1$ and $S_2$ is finite. In particular, $\wh S=
\wh{\overline S}$.

It is convenient to consider the affine real line $\A^1(R)=R$ as
contained in the projective real line $\P^1(R)=R\cup\{\infty\}$. The
semi-algebraic subsets of $\P^1(R)$ are the subsets which intersect
$R$ in a semi-algebraic set. For any such set $S$ we define $\wh S$
to be $(S-\{\infty\})\whpx$.

We remark that the above description of the constructible sets in
$X_{R(t)}$ via \sa\ sets extends naturally to more general
situations. In particular, given any integral algebraic variety $V$
over $R$ and any \sa\ subset $S$ of $V(R)$, one defines an
open-closed subset $\wh S$ of $X_{R(V)}$ in a similar and natural
way. (See any of the standard references like \cite{BCR}, \cite{KS}
or \cite{PD}.) The open-closed subsets of $X_{R(V)}$ are precisely
these sets $\wh S$, and $\wh S_1=\wh S_2$ holds if and only if 
the set-theoretic
difference of $S_1$ and $S_2$ is not Zariski dense in $V$.

For the purpose of this appendix, only the case $V=\P^1$ is needed, as
explained above. The notation $\wh S$ is not standard and has been
chosen \emph{ad~hoc} here.

We denote the topological (disjoint) sum of two topological spaces
$X$ and $Y$ by $X\amalg Y$, and write $X\approx Y$ to indicate that
$X$ and $Y$ are homeomorphic.

\begin{lemnum}\label{lema}
Let $I_1,\dots,I_r$ ($r\ge1$) be any finite number of non-degenerate
intervals in $R$. Then $X_{R(t)}\approx\wh I_1\amalg\cdots\amalg
\wh I_r$.
\end{lemnum}

Here, an interval $I$ in $R$ is called non-degenerate if it contains
more than one point.

\begin{proof}
It is convenient to extend the concept of intervals from $R$ to $\P^1
(R)$ in the obvious way. For the proof of the lemma we may restrict
to closed intervals. %,to fix ideas. 
Any two nonempty closed intervals
in $\P^1(R)$, different from $\P^1(R)$, are conjugate under the
action of $\PGL_2(R)$. From this it follows that $\wh I_1\approx
\wh I_2$ holds for any two such intervals $I_1$, $I_2$, and also that
$\wh I\approx X_{R(t)}-\wh I$ for any such interval $I$. Since
clearly $[-1,1]\whpx\approx[-1,0]\whpx\amalg[0,1]\whpx$, and since
$X_{R(t)}\approx[0,1]\whpx\amalg\bigl(X_{R(t)}-[0,1]\whpx\bigr)$, the
assertion of the lemma follows.
\end{proof}

\begin{propnum}\label{saz}
Let $R\subset K$ be a finitely generated field extension of
trans\-cendence degree one. If the field $K$ is real then $X_K$ is
homeomorphic to $X_{R(t)}$.
\end{propnum}

\begin{proof}
Choose $t$, $x\in K$ with $t$ transcendental over $R$ and $K=
R(t,x)$, and let $r\colon X_K\to X_{R(t)}$ be the restriction map.
The discriminant of the minimal polynomial of $x$ over $R(t)$ is a
nonzero element of $R(t)$. Its real poles and zeros, together with
$\infty$, divide the real $t$-line into finitely many open intervals
$I_1,\dots,I_r$. For each index $i=1,\dots,r$, the restriction
$r^{-1}(\wh I_i)\to\wh I_i$ of $r$ is topologically a trivial
$d_i$-sheeted covering, where $0\le d_i\le[K:R(t)]$ is some integer.
Therefore, $X_K$ is homeomorphic to a topological sum $\coprod_{i=1}
^r\coprod_{j=1}^{d_i}\wh I_i$, and so the above Lemma shows that
$X_K$ is homeomorphic to $X_{R(t)}$.
\end{proof}

\begin{cornum}
Let $R\subset K$ be a finitely generated field extension of
transcendence degree one. Given any finite number of nonempty
open-closed subsets $U_1,\dots,U_r$ of $X_K$ ($r\ge1$), there is a
homeomorphism $X_{R(t)}\approx U_1\amalg\cdots\amalg U_r$.
\end{cornum}

\begin{proof}
This follows from a combination of the two previous results.
\end{proof}\vspace*{0.3cm}

\noindent {\large \bf Acknowledgements}\\

\noindent The authors wish to acknowledge financial 
support provided by the European RTN Network 
\emph{Algebraic $K$-Theory, Linear Algebraic Groups and Related Structures} (HPRN
CT-2002-0287). They are grateful to Claus Scheiderer for contributing 
the Appendix and for helpful comments that allowed to shorten the exposition in
section 3.1.

\bibliographystyle{amsplain}
\begin{thebibliography}{99}

\bibitem{BCR} Bochnak J., Coste M., Roy M.-F.:
\emph{Real Algebraic Geometry}.
Erg.\ Math.\ Grenzgeb.\ (3) \textbf{36}, Springer, Berlin, 1998.

\bibitem{craven} Craven T.: The topological space of orderings of a rational function field,
Duke Math. J. {\bf 41} (1974), 339--347.

\bibitem{elp} Elman R., Lam T. Y., Prestel A.: On some Hasse Principles over Formally Real Fields, Math.
Z. \textbf{134} (1973), 291-301.

\bibitem{gb} Grenier-Boley N.: Harrison's criterion, Witt equivalence and reciprocity
equivalence, to appear in the Bulletin of the Belgian Mathematical Society-Simon Stevin.

\bibitem{harrison} Harrison D. K.: \emph{Witt rings}, University of Kentucky
Notes, Lexington, Kentucky, 1970.

\bibitem{KS} Knebusch M., Scheiderer C.: \emph{Einf\"uhrung in die reelle Algebra}.
Vieweg, Wiesbaden, 1989.

\bibitem{koprowski} Koprowski P.: Witt equivalence of algebraic function fields over real closed fields,
Math. Z. {\bf 242} (2002), $\mathrm{n^o}$ 2, 323--345.

\bibitem{lam1} Lam T. Y.: \emph{Orderings, Valuations, and Quadratic
Forms}, CBMS Regional Conference Series in Mathematics, \textbf{52}.
American Mathematical Society, Providence, RI, 1983.

\bibitem{lam2} Lam T. Y.: \emph{Introduction to quadratic forms over fields}, 
Graduate Studies in Mathematics, \textbf{67}. American Mathematical Society,
Providence, RI, 2005.

\bibitem{pscl} Perlis R., Szymiczek K., Conner P. E., Litherland R.: Matching Witts with global fields,
Contemp. Math.  \textbf{155} (1994), 365--387.

\bibitem{pfis} Pfister A.: \emph{Quadratic Forms
with Applications to Algebraic Geometry and Topology},
London Math.\ Soc.\ Lecture Notes Series \textbf{217}, 
Cmbridge University Press 1995.

\bibitem{PD} Prestel A., Delzell C.: \emph{Positive Polynomials}.
Monographs in Mathematics, Springer, Berlin, 2001.

\bibitem{scharlau} Scharlau W.: \emph
{Quadratic and Hermitian Forms}, Grundlehren Math. Wiss.
\textbf{270}, Berlin, Springer-Verlag 1985.

\bibitem{stone} Stone, M.H.: \emph{Applications of the theory of 
Boolean rings to general topology}, Trans.\ Amer.\ Math.\ Soc.\   
\textbf{41} (1937),  $\mathrm{n^o}$ 3, 375--481
\end{thebibliography}

\noindent{\sc IUFM de Rouen, 2 rue du Tronquet, BP 18, 76131 Mont-Saint-Aignan Cedex, France}\\
{\it E-mail address:} {\tt nicolas.grenier-boley@univ-rouen.fr}\\

\noindent{\sc Division of Pure Mathematics, School of
Mathematical Sciences, University of
Nottingham, University Park, Nottingham NG7 2RD, UK}\\
{\it E-mail address:} {\tt Detlev.Hoffmann@nottingham.ac.uk}\\

\noindent{\sc Universit\"at Konstanz, Fachbereich Mathematik und Statistik, 
78457 Konstanz, Germany}\\
{\it E-mail address:} {\tt claus.scheiderer@uni-konstanz.de}


\end{document}
