%%% % 24 May 2004
\NeedsTeXFormat{LaTeX2e}
\documentclass[a4paper,leqno,11pt,final]{amsproc}
\usepackage{amssymb,latexsym,amsxtra,amscd,amsfonts,amsthm,amsmath,verbatim}

\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\Br}{Br}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\cor}{cor}
\DeclareMathOperator{\iso}{\stackrel{\sim}{\to}}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\charac}{char}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\sgn}{sgn}

\newcommand{\RS}{\mathfrak{R}}
\newcommand{\invo}{\mkern1mu\rule[1.3ex]{0.48em}{0.1ex}\mkern1mu}
\newcommand{\half}{{\textstyle\frac12}}

\newcommand{\PP}{\mathbb{P}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\Aff}{\mathbb{A}}

\newcommand{\Sym}{\mathfrak{S}}

\renewcommand{\theequation}{\arabic{equation}}
\numberwithin{equation}{section}


\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{corol}[thm]{Corollary}
\newtheorem*{lem}{Lemma}

\theoremstyle{definition}
\newtheorem*{defn}{Definition}

\theoremstyle{remark}
\newtheorem*{remark}{Remark}

\textwidth = 12cm
\textheight = 18cm

\title[Division Algebras Over Function Fields]{Division
Algebras Over Rational Function Fields in One Variable}

\author{L.H.~Rowen}
\address{Bar-Ilan University\\
Ramat Gan, 52100 Israel}
\email{rowen@macs.biu.ac.il}

\author{A.S.~Sivatski}
\address{St Petersburg Electrotechnical University\\
197376 St Petersburg, Russia}
\email{sivatsky@as3476.spb.edu}

\author{J.-P.~Tignol}
\thanks{Work supported in part by the European Community's Human
Potential Programme under contract HPRN--CT--2002--00287, KTAGS.
Research of the first author was funded in part by an Israel Academy
of Sciences Center of Excellence.
The second author was supported in part by INTAS under contracts
00--566 and 99--00817. He gratefully acknowledges the hospitality of
the Institut de Math\'ematique Pure et Appliqu\'ee, Universit\'e
catholique de Louvain, where research leading to this paper was
initiated. The
third author is supported in part by the National Fund for Scientific
Research (Belgium).}
\address{Universit\'e catholique de Louvain\\
1348 Louvain-la-Neuve, Belgium}
\email{tignol@math.ucl.ac.be}
\urladdr{http://www.math.ucl.ac.be/tignol}

\begin{document}

\keywords{Division algebra, Brauer group, rational functions, ramification}
\subjclass[2000]{16K20, 14H05}

\renewcommand*{\abstractname}{{\upshape\textbf{Abstract}}}
\begin{abstract}
   Let $A$ be a central simple algebra over the field of rational
   functions in one variable over an arbitrary field of characteristic
   different from~$2$. If the Schur index of $A$ is not divisible by
   the characteristic and its ramification locus has degree at most
   $3$, then $A$ is Brauer-equivalent to the tensor product of a
   quaternion algebra and a constant central division algebra $D$. The
   index of $A$ is computed in terms of $D$ and the ramification of $A$.
   The result is used to construct various examples of division
   algebras over rational function fields.
\end{abstract}

\maketitle

\section*{Introduction}

The Brauer group of the field of rational fractions in one variable
over an arbitrary field $F$ of characteristic zero has been described
by D.K.~Faddeev in \cite{Fad}. To recall his result, we use the
following notation: for any closed point $p$ of the projective line
$\PP^{1}_{F}$, let $F_{p}$ denote the residue field at $p$. Let
$X(F_{p})$ be the character group of the absolute Galois group
$\Gamma_{p}$ of $F_{p}$, i.e.\ the group of continuous homomorphisms
\[
X(F_{p})=\Hom(\Gamma_{p},\QQ/\ZZ).
\]
For each closed point $p$ on $\PP^{1}_{F}$, there is a map
$\partial_{p}\colon\,\Br F(t)\to X(F_{p})$ known as the
\emph{ramification} (or \emph{residue}) map at $p$ (see \cite[p.~18]{GMS}).
Faddeev proved that the
following sequence is exact:
\begin{equation}
     \label{Fad.eq}
0\to\Br F\to \Br F(t)
\xrightarrow{\oplus\partial_{p}}
\bigoplus_{p\in\PP^{1(1)}_{F}}
X(F_{p})
\xrightarrow{\sum\cor}
X(F)\to 0,
\end{equation}
where $\sum\cor$ is the sum of the corestriction (or norm) maps
$X(F_{p})\to X(F)$ and $\PP^{1(1)}_{F}$ is the set of closed points
on $\PP^{1}_{F}$. See also \cite[Example~9.21, p.~26]{GMS}, where a
version in nonzero characteristic is given. If the characteristic of
$F$ is $p$, Faddeev's exact sequence still holds if the Brauer groups
and the character groups are replaced by their subgroups of
prime-to-$p$ torsion.
\medbreak
\par
Our aim is to obtain information on the index of Brauer classes in
$\Br F(t)$ from their image under the ramification map
$\partial=\oplus\partial_{p}$. We achieve this goal in some very
specific cases. To describe them, we use the following terminology:

\begin{defn}
     Let
     \[
     \RS=\Ker\Bigl(
     \sum\cor\colon\,\bigoplus_{p\in\PP^{1(1)}_{F}}X(F_{p})\to X(F)
     \Bigr).
     \]
     The elements in $\RS$ are called \emph{ramification sequences}.
     For $\rho=(\chi_{p})\in\RS$, the \emph{support}
     of $\rho$ is
     \[
     \supp(\rho)=\{p\in\PP^{1(1)}_{F}\mid\chi_{p}\neq0\},
     \]
     and $\chi_{p}$ is the \emph{component} of $\rho$ at $p$.
     Viewing $\supp(\rho)$ as a divisor on $\PP^{1}_{F}$, we may consider
     the \emph{degree} of the support of $\rho$:
     \[
     \deg\supp(\rho)=\sum_{p\in\supp(\rho)}\deg p.
     \]
\end{defn}

If $p$ is a rational point (i.e., $F_{p}=F$), then
$\cor\colon\,X(F_{p})\to X(F)$ is the identity map. Therefore, the
support of a ramification sequence cannot consist of a single rational
point,
\[
\deg\supp(\rho)\geq2\qquad\text{for all $\rho\in\RS$}.
\]

In this work, we consider only $2$-torsion ramification sequences (and
may therefore assume only $\charac F\neq2$).
Their set is denoted by ${}_{2}\RS$. For $\rho\in{}_{2}\RS$ with
$\deg\supp(\rho)=2$, we determine in Corollaries~\ref{ramloc21.cor}
and \ref{ramloc22.cor} the quaternion $F(t)$-algebras $Q$ such that
$\partial(Q)=\rho$. If $\alpha\in\Br F(t)$ has torsion prime to $\charac
F$ and the same ramification
sequence $\rho$, then by Faddev's exact
sequence \eqref{Fad.eq} there is a
central simple $F$-algebra $A$ and a quaternion $F(t)$-algebra $Q$
such that $\alpha$ is the Brauer class of $A\otimes_{F}Q$,
\[
\alpha=[A\otimes_{F}Q]\in\Br F(t).
\]
The index of $\alpha$ is determined in terms of $A$ and $\rho$ in
Theorem~\ref{locram2ind.thm}.

In Section~\ref{ramloc3.sec}, we obtain similar results for
$\rho\in{}_{2}\RS$ with $\deg\supp(\rho)=3$. In this case, we show in
Corollary~\ref{ramloc3.cor} that there is up to isomorphism a unique
quaternion $F(t)$-algebra $Q_{\rho}$ such that
\[
\partial(Q_{\rho})=\rho.
\]
For any central simple $F$-algebra $A$, the index of
$A\otimes_{F}Q_{\rho}$ is computed in Theorem~\ref{locram3ind.thm} in
terms of $A$ and a quartic field extension $E_{\rho}/F$ canonically
associated with $\rho$.

The existence of quaternion $F(t)$-algebras with given $2$-torsion
ramification sequence $\rho$ with support of degree at most $3$ is
also shown in a recent paper of
Kunyavski\u\i, Rowen, Tikhonov and Yanchevski\u\i\
\cite[Corollary~2.10]{KRTY}, except in the case where the support consists
of a
single point of degree~$3$. Ramification sequences of quaternion
algebras are further discussed in \cite{KRTY}, which also gives
(in Corollary~2.14) an example due to Faddeev of a ramification sequence
$\rho\in{}_{2}\RS$ with $\deg\supp(\rho)=4$ which is not the ramification
sequence of a quaternion $F(t)$-algebra.

The main ingredients in the proofs of our main results are a very
general observation on cochains with values in a left principal ideal
domain (Section~\ref{coch.sec}) and a reduction of $2$-torsion
ramification sequences $\rho$ with $\deg\supp(\rho)\leq3$ to a normal
form, which is achieved in Section~\ref{ramseq.sec}. Although the
restriction to $2$-torsion elements in $\RS$ is not necessary for
some of the reduction steps in Section~\ref{ramseq.sec}, an example
given in an appendix to this paper shows that even for $3$-torsion
ramification sequences the general principles of
Section~\ref{coch.sec} lead to conditions which are much more
difficult to handle.

\section{Cochains in the ring of fractions of a left PID}
\label{coch.sec}

This section presents the basic tool that will be used in subsequent
sections to prove that certain tensor products are division algebras.
As it is very general, we cast it in the setting of left Principal
Ideal Domains (PID) although we apply it only in the case of
polynomial rings in one indeterminate over division rings.
\medbreak
\par
Let $R$ be a left PID, with Ore ring of fractions $Q$,
\[
Q=\{b^{-1}a\mid a,b\in R, b\neq0\}.
\]
Let $G$ be a finite group of automorphisms of $R$, that we extend to
$Q$.

\begin{lem}
     Suppose $(c_{\sigma})_{\sigma\in G}$ is a family of elements in
     $Q^\times$ such that
     \[
     c_{\sigma\tau}^{-1}\sigma(c_{\tau})c_{\sigma}\in R
     \qquad\text{for all $\sigma$, $\tau\in G$}.
     \]
     There exists $q\in Q^\times$ such that
     \[
     \sigma(q)c_{\sigma}q^{-1}\in R
     \qquad\text{for all $\sigma\in G$}.
     \]
\end{lem}

\begin{proof}
     \relax From the definition of $Q$, it follows that for each
     $\sigma\in G$ the left ideal
     \[
     I_{\sigma}=\{g\in R\mid \sigma(g)c_{\sigma}\in R\}
     \]
     is nonzero. As $R$ is left Ore (as it is a left PID) and $G$ is
     finite, the intersection
     \[
     I=\bigcap_{\sigma\in G}I_{\sigma}
     =\{g\in R\mid\sigma(g)c_{\sigma}\in R
     \text{ for all $\sigma\in G$}\}
     \]
     is a nonzero left ideal. Let $q\in R$ be such that
     \[
     I=Rq,
     \]
     hence $q\neq0$. For all $\tau\in G$ we have $\tau(q)c_{\tau}\in
     R$ since $q\in I$. Moreover, for $\sigma\in G$,
     \[
     \sigma\bigl(\tau(q)c_{\tau}\bigr)c_{\sigma}=
     \sigma\tau(q)\sigma(c_{\tau})c_{\sigma}=
     \bigl(\sigma\tau(q)c_{\sigma\tau}\bigr)
     \bigl(c_{\sigma\tau}^{-1}\sigma(c_{\tau})c_{\sigma}\bigr).
     \]
     The first factor on the right side lies in $R$ because $q\in I$,
     and the second factor lies in $R$ by hypothesis. Therefore,
     \[
     \sigma\bigl(\tau(q)c_{\tau}\bigr)c_{\sigma}\in R
     \qquad\text{for all $\sigma$, $\tau\in G$},
     \]
     hence
     \[
     \tau(q)c_{\tau}\in I\qquad\text{for all $\tau\in G$},
     \]
     and therefore $\tau(q)c_{\tau}\in Rq$ for all $\tau\in G$.
\end{proof}

The only application of this lemma in the present paper is the
following: Let $E$ be a (finite-dimensional)
central division algebra over an arbitrary
field $K$, and let $x$ be an indeterminate over $K$. The ring
$E[x]=E\otimes_{K}K[x]$ is a left (and right) PID with Ore ring of
fractions $E(x)=E\otimes_{K}K(x)$. Let $\alpha$ be an automorphism
of finite order $n$ of $E[x]$, and let $g\in K[x]$. Consider the
algebra
\begin{equation}
     \label{Delta.eq}
\Delta(E(x),\alpha,g)=E(x)\oplus E(x)y\oplus\cdots\oplus
E(x)y^{n-1}
\end{equation}
where multiplication is defined by
\[
y^n=g\qquad\text{and}\qquad
yf=\alpha(f)y\text{ for $f\in E(x)$}.
\]
This algebra may be viewed as the factor ring of the skew-polynomial
ring $E(x)[y;\alpha]$ by the ideal generated by $y^n-g$.

\begin{thm}
     \label{Delta.thm}
     Suppose $n$ is a prime and $\alpha$ does not restrict to the
     identity on $K(x)$. The algebra $\Delta(E(x),\alpha,g)$ is a
     division algebra if and only if there is no $f\in E[x]$ such that
     \begin{equation}
	\label{norm.eq}
	\alpha^{n-1}(f)\ldots\alpha(f)f=g.
     \end{equation}
\end{thm}

\begin{proof}
     By \cite[Theorem~1.3.16, p.~17]{Jake} (see also
     \cite[Theorem~11.12, p.~184]{SofA}), the algebra
     $\Delta(E(x),\alpha,g)$ is not division if and only if there
     exists $f\in E(x)$ for which \eqref{norm.eq} holds. Therefore, it
     suffices to show that if \eqref{norm.eq} holds for some $f\in
     E(x)$, then it also holds for some $f\in E[x]$.

     Suppose $f\in E(x)^\times$ satisfies \eqref{norm.eq}, and let
     \[
     c_{\alpha^i}=\alpha^{i-1}(f)\ldots\alpha(f)f
     \qquad\text{for $i=0$, \ldots, $n-1$}.
     \]
     Then, for $i$, $j=0$, \ldots, $n-1$,
     \[
     c_{\alpha^{i+j}}^{-1}\alpha^i(c_{\alpha^j})c_{\alpha^i}=
     \begin{cases}
	1&\text{if $i+j<n$},\\
	g&\text{if $i+j\geq n$}.
     \end{cases}
     \]
     We may therefore apply the lemma to find $q\in E(x)^\times$ such
     that
     \[
     \alpha(q)c_{\alpha}q^{-1}\in E[x].
     \]
     Let $f'=\alpha(q)c_{\alpha}q^{-1}=\alpha(q)fq^{-1}$. We have
     \[
     \alpha^{n-1}(f')\ldots\alpha(f')f'=qgq^{-1}=g,
     \]
     hence $f'\in E[x]$ satisfies \eqref{norm.eq}.
\end{proof}

We consider two types of examples, where
$\Delta(E(x),\alpha,g)$ turns out to be the tensor product of a
``constant'' division algebra and a quaternion algebra. One more
example is discussed in the Appendix.

\subsection{Quaternion algebras with a constant slot}

Let $D$ be a central division algebra over a field $F$ of
characteristic different from~$2$, and let $a\in F^\times$ be such
that $D$ does not contain a square root of $a$. Then
$E=D\otimes_{F}F(\sqrt{a})$ is a central division algebra over
$K=F(\sqrt{a})$, and the nontrivial automorphism of $K/F$ extends to
an automorphism $\alpha$ of $E$ which is the identity on $D$. Let $t$
be an indeterminate over $F$ and $g\in F[t]$. The algebra
$\Delta(E(t),\alpha,g)$ of \eqref{Delta.eq} is a tensor product:
\[
\Delta(E(t),\alpha,g)=D\otimes_{F}(a,g)_{F(t)}.
\]
Theorem~\ref{Delta.thm} shows that this tensor product has zero
divisors if and only if there exists $f\in E[t]$ such that
\[
\alpha(f)f=g.
\]
Since the degree of the left side is even, the following result is
clear:

\begin{corol}
     Let $D$ be a central division $F$-algebra which does not contain a
     square root of $a$. For every $g\in F[t]$ of odd degree, the
     tensor product
     $D\otimes_{F}(a,g)_{F(t)}$ is a division algebra.
\end{corol}

We now consider a case where $\deg g=2$.

\begin{prop}
     \label{degg2.prop}
     Let $D$ be a central division $F$-algebra which does not contain
     a square root of $a$, and let $b\in F^\times$. The tensor product
     $D\otimes_{F}(a,t^{2}-b)_{F(t)}$ is not a division algebra if and
     only if $D$ contains an element $s$ such that $s^{2}=ab$.
\end{prop}

\begin{proof}
     As observed in the beginning of this subsection, the tensor product
     $D\otimes_{F}(a,t^{2}-b)_{F(t)}$ is not a division algebra if and
     only if there exists $f\in E[t]$ such that
     \begin{equation}
	\label{degg2.eq}
     \alpha(f)f=t^{2}-b.
     \end{equation}
     Comparing degrees, we see that if $f$ exists it must be of the
     form
     \[
     f=ut+v\qquad\text{for some $u$, $v\in E$}.
     \]
     Substituting in \eqref{degg2.eq}, we obtain
     \begin{equation}
	\label{degg22.eq}
	\alpha(u)u=1,\qquad
	\alpha(u)v+\alpha(v)u=0,\qquad
	\alpha(v)v=-b.
     \end{equation}
     Letting $x=\sqrt{a}\alpha(u)v$, computation shows
     that the existence of $u$, $v\in E$ satisfying \eqref{degg22.eq}
     is equivalent to the existence of $u$, $x\in E$ satisfying
     \begin{equation}
	\label{degg23.eq}
	\alpha(u)u=1,\qquad
	\alpha(x)=uxu^{-1},\qquad
	x^{2}=ab.
     \end{equation}

     If $D$ contains $s$ such that $s^{2}=ab$, then \eqref{degg23.eq}
     holds with $u=1$ and $x=s$.

     Conversely, suppose \eqref{degg23.eq} holds for some $u$, $x\in E$.
     If $u=1$, then $\alpha(x)=x$ hence $x\in D$ and we may set $s=x$.
     If $u\neq1$, then letting $w=u-1\in E^\times$ we have
     \[
     \alpha(w)u=1-u=-w.
     \]
     For $s=wxw^{-1}$, it follows that
     \[
     \alpha(s)=\alpha(w)\alpha(x)\alpha(w)^{-1}=
     \alpha(w)uxu^{-1}\alpha(w)^{-1}=wxw^{-1}=s,
     \]
     so $s\in D$, and $s^{2}=ab$.
\end{proof}

\subsection{Quaternion algebras with an indeterminate slot}
\label{indetslot.subsec}

Let $D$ be a central division algebra over a field $F$ of
characteristic different from~$2$ and let $t$ be an indeterminate
over $F$. For $g\in F[t]$, we may consider the tensor product
$D\otimes_{F}(t,g)_{F(t)}$ as a special case of the construction in
\eqref{Delta.eq}:
\[
D\otimes_{F}(t,g)_{F(t)}=\Delta\bigl(D(x),\alpha,g(x^{2})\bigr)
\]
where $x^{2}=t$ and $\alpha$ is the automorphism of $D(x)$ defined by
\[
\alpha(x)=-x
\qquad\text{and}\qquad
\alpha(d)=d\text{ for $d\in D$}.
\]
Therefore, Theorem~\ref{Delta.thm} shows that this tensor
product is not a division algebra if and only if there exists $f\in
D[x]$ such that
\begin{equation}
     \label{gxsquare.eq}
\alpha(f)f=g(x^{2}).
\end{equation}
We just consider one example where $\deg g=3$.

\begin{prop}
     \label{degg3.prop}
     Let $D$ be a central division $F$-algebra and $a$, $b$, $c\in F$.
     The tensor product
     \[
     D\otimes_{F}(t,a^{2}-bt+ct^{2}-t^{3})_{F(t)}
     \]
     is not a division algebra if and only if $D$ contains an element
     $s$ such that
     \begin{equation}
	\label{degg3.eq}
	s^{4}-2cs^{2}-8as+(c^{2}-4b)=0.
     \end{equation}
\end{prop}

\begin{proof}
     Let $g=a^{2}-bt+ct^{2}-t^{3}\in F[t]$. If $s\in D$ satisfies
     \eqref{degg3.eq}, then the polynomial
     \[
     f=a+\half(s^{2}-c)x+sx^{2}+x^{3}
     \]
     satisfies $\alpha(f)f=g(x^{2})$, hence, by the observations at
     the beginning of this subsection, the tensor product
     $D\otimes(t,g)_{F(t)}$ is not a division algebra.

     Conversely, if $D\otimes(t,g)_{F(t)}$ is not a division algebra,
     then \eqref{gxsquare.eq} holds for some $f\in D[x]$, which is
     necessarily of the form
     \[
     f=rx^{3}+sx^{2}+ux+v
     \qquad\text{for some $r$, $s$, $u$, $v\in D$}.
     \]
     Comparing the leading coefficients and the constant terms in
     \eqref{gxsquare.eq} yields
     \[
     r^{2}=1\qquad\text{and}\qquad
     v^{2}=a^{2}.
     \]
     Changing the sign of $f$ and/or changing $f$ into $\alpha(f)$ if
     necessary, we may assume $r=1$ and $v=a$. Expanding
     \eqref{gxsquare.eq} then yields the following relations between
     $s$ and $u$:
     \[
     s^{2}-2u=c,\qquad
     u^{2}-2as=b.
     \]
     Eliminating $u$, we obtain \eqref{degg3.eq}.
\end{proof}

\begin{remark}
     The cubic resolvent of the quartic equation \eqref{degg3.eq} is
     $g(t)=0$. The conceptual relation between $g$ and the quartic
     polynomial in
     \eqref{degg3.eq} will be made explicit in
     Section~\ref{ramloc3.sec}.
\end{remark}

\section{Transformations of the projective line}

A choice of projective coordinates in the projective line
$\PP^{1}_{F}$ over an arbitrary field $F$ is an identification
\[
\PP^{1}_{F}=\Proj(F[u,v])
\]
where $u$, $v$ are indeterminates of degree~$1$ over $F$. When
projective coordinates are chosen, the rational points of
$\PP^{1}_{F}$ are identified with $F\cup\{\infty\}$ in such a way
that $a\in F$ corresponds to the homogeneous ideal $(u-av)F[u,v]$,
and $\infty$ to $vF[u,v]$. We then identify the field of rational
functions on $\PP^{1}_{F}$ with $F(t)$, where $t=uv^{-1}$, and
$\PP^{1}_{F}\setminus\{\infty\}=\Aff^{1}_{F}=\Spec(F[t])$.

It is well-known that the group of transformations of $\PP^1_{F}$
is simply transitive on the triples
of rational points, see for instance \cite[\S~III.3]{Baer}.
In this section, we
prove an analogous result for divisors consisting
of a single point of degree~$3$, and for divisors consisting of a rational
point and a point of degree~$2$.
\par
For clarity, we treat separately the various cases, although a uniform
proof should be possible.

\begin{prop}
     \label{1+2.prop}
     Let $p$, $p'$ be rational points on $\PP^{1}_{F}$ and $q$, $q'$
     be closed points of degree~$2$. The projective transformations
     of $\PP^{1}_{F}$ which map $p$ to $p'$ and $q$ to $q'$ are in
     one-to-one correspondence with the $F$-isomorphisms $F_{q}\iso
     F_{q'}$.
\end{prop}

\begin{proof}
     After a projective transformation, we may assume $p=p'$ and choose
     projective coordinates such that $p$ is the point at infinity. The
     projective transformations which leave $p$ invariant then are the
     transformations of the affine line $\Aff^{1}_{F}$.

     Viewing $q$ and $q'$ as closed points on
     $\Aff^{1}_{F}=\Spec(F[t])$, consider monic irreducible
     polynomials $\pi$, $\pi'\in F[t]$ of degree~$2$ such that
     \[
     q=\pi F[t]
     \qquad\text{and}\qquad
     q'=\pi'F[t].
     \]
     The affine
     transformation $t\mapsto at+b$ (with $a$, $b\in F$, $a\neq0$) maps $q$ to
     $q'$ if and only if
     \begin{equation}
	\label{aff.eq}
     \pi(at+b) F[t]=\pi'(t) F[t].
     \end{equation}
     Let $\tau'\in F_{q'}$ be the image of $t$ under the canonical
     epimorphism $F[t]\to F[t]/(\pi')=F_{q'}$. Equation~\eqref{aff.eq}
     holds if and only if $\pi$ is the minimal polynomial of
     $a\tau'+b$ over $F$. Since every element in $F_{q'}$ has the form
     $a\tau'+b$
     for some $a$, $b\in F$, affine transformations which map $q$ to $q'$
     are in
     one-to-one correspondence with the elements in $F_{q'}$ whose
     minimal polynomial is $\pi$, hence also with the $F$-isomorphisms
     $F_{q}\iso F_{q'}$.
\end{proof}

\begin{prop}
     \label{3.prop}
     Let $r$, $r'$ be closed points of degree~$3$ on $\PP^{1}_{F}$. The
     projective transformations of $\PP^{1}_{F}$ which map $r$ to $r'$
     are in one-to-one correspondence with the $F$-isomorphisms
     $F_{r}\iso F_{r'}$.
\end{prop}

\begin{proof}
     Choose coordinates to represent $\PP^{1}_{F}$ as
     $\Proj(F[u,v])$, where $u$, $v$ are indeterminates of degree~$1$.
     We may then find homogeneous irreducible polynomials $\pi$,
     $\pi'\in F[u,v]$ of degree~$3$ such that
     \[
     r=\pi F[u,v]
     \qquad\text{and}\qquad
     r'=\pi'F[u,v].
     \]
     After scaling, we may assume the coefficients of $u^{3}$ in $\pi$
     and $\pi'$ are $1$. The projective transformation $u\mapsto
     au+bv$, $v\mapsto cu+dv$ (where $a$, $b$, $c$, $d\in F$ and
     $ad-bc\neq0$) maps $r$ to $r'$ if and only if
     \[
     \pi(au+bv,cu+dv)F[u,v]=\pi'(u,v)F[u,v]
     \]
     or, equivalently (after de-homogeneizing),
     \begin{equation}
	\label{3.eq}
	\pi(at+b,ct+d)F[t]=\pi'(t,1)F[t].
     \end{equation}
     Let $\tau'\in F_{r'}$ be the image of $t$ under the canonical
     epimorphism $F[t]\to F[t]/\bigl(\pi'(t,1)\bigr)=F_{r'}$.
     Equation~\eqref{3.eq} holds if and only if
     $\pi(a\tau'+b,c\tau'+d)=0$, which means that $\pi(t,1)$ is the
     minimal polynomial of $\frac{a\tau'+b}{c\tau'+d}$. As in the proof
     of
     Proposition~\ref{1+2.prop}, it now suffices to establish the
     following result:
     \medbreak
     \par\noindent
     \textit{Claim:}
     For every $x\in F_{r'}$, $x\notin F$, there exist $a$, $b$, $c$,
     $d\in F$, uniquely determined up to a scalar factor, such that
     $x=\frac{a\tau'+b}{c\tau'+d}$ and $ad-bc\neq0$.
     \medbreak
     \par
     Since $\tau'\notin F$ and $x\neq0$, we have
     \[
     \dim_{F}(F\tau'+F)=\dim_{F}x(F\tau'+F)=2,
     \]
     hence, since $\dim_{F}F_{r'}=3$,
     \[
     \dim_{F}\bigl((F\tau'+F)\cap x(F\tau'+F)\bigr)\geq1.
     \]
     If the inequality is strict, then
     \begin{equation}
	\label{3.1.eq}
	x(F\tau'+F)=F\tau'+F,
     \end{equation}
     hence $x\in F\tau'+F$. Since $x\notin F$, it follows that
     $Fx+F=F\tau'+F$, hence $\tau'\in Fx+F$ and \eqref{3.1.eq} implies
     that $F\tau'+F$ is a subalgebra of $F_{r'}$. This is a
     contradiction since $\tau'$ has degree~$3$. Therefore,
     \[
     \dim_{F}\bigl((F\tau'+F)\cap x(F\tau'+F)\bigr)=1.
     \]
     We may thus find $a$, $b$, $c$, $d\in F$, uniquely determined up
     to a scalar, such that
     $
     a\tau'+b=x(c\tau'+d)\neq0
     $, hence
     \[
     x=\frac{a\tau'+b}{c\tau'+d}.
     \]
     If $ad-bc=0$, then $x\in F$, a
     contradiction.
\end{proof}

Viewing projective transformations as changes of projective
coordinates, we readily deduce from Propositions~\ref{1+2.prop} and
\ref{3.prop} (and from the ``second fundamental theorem of projective
geometry'' \cite[\S~III.3]{Baer}):

\begin{corol}
     \label{projco.cor}
     \begin{enumerate}
	\item
	Let $p_{1}$, $p_{2}$, $p_{3}\in\PP^{1(1)}_{F}$ be distinct
	rational points and
	let $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}\in F[u,v]$ be
	homogeneous polynomials of degree~$1$ which are pairwise distinct up
	to scalars. There is a choice of projective coordinates
	$\PP^{1}_{F}=\Proj(F[u,v])$ such that
	\[
	p_{1}=\lambda_{1}F[u,v],
	\quad
	p_{2}=\lambda_{2}F[u,v],
	\quad
	p_{3}=\lambda_{3}F[u,v].
	\]
	\item
	Let $p\in \PP^{1(1)}_{F}$ be a rational point and
	$q\in \PP^{1(1)}_{F}$ be
	a point of degree~$2$. Let $\lambda\in F[u,v]$ be a
	homogeneous polynomial of degree~$1$ and $\pi\in F[u,v]$ be a
	homogeneous polynomial of degree~$2$ such that $F_{q}\simeq
	F[t]/\bigl(\pi(t,1)\bigr)$. There is a choice of projective
	coordinates $\PP^{1}_{F}=\Proj(F[u,v])$ such that
	\[
	p=\lambda F[u,v],
	\qquad
	q=\pi F[u,v].
	\]
	\item
	Let $r\in \PP^{1(1)}_{F}$ be a point of degree~$3$ and $\pi\in
	F[u,v]$ be a homogeneous polynomial of degree~$3$ such that
	$F_{r}\simeq F[t]/\bigl(\pi(t,1)\bigr)$. There is a choice of
	projective coordinates $\PP^{1}_{F}=\Proj(F[u,v])$ such that
	\[
	r=\pi F[u,v].
	\]
     \end{enumerate}
\end{corol}

\section{$2$-torsion ramification sequences}
\label{ramseq.sec}

In this section, we assume that the characteristic of the base field
$F$ is different from~$2$. We use projective transformations to set
$2$-torsion
ramification sequences with support of degree at most~$3$ into a
standard form.

Let $\Gamma$ be the absolute Galois group of $F$ and
$\mu_{2}=\{\pm1\}\subset F^\times$. Since $\charac F\neq2$, we may
identify the $2$-torsion characters of $\Gamma$ with square classes
of $F$, via the isomorphisms
\[
{}_{2}X(F)=\Hom(\Gamma,\ZZ/2\ZZ)=H^{1}(\Gamma,\mu_{2})=F^\times/F^{\times2}.
\]
A ramification sequence $\rho\in{}_{2}\RS$ can therefore be viewed as an
element in the kernel of the norm map
\begin{equation}
     \label{N.eq}
     \bigoplus_{p\in\PP^{1(1)}_{F}}F^\times_{p}/F_{p}^{\times2}\to
     F^\times/F^{\times2}.
\end{equation}

Suppose first $\deg\supp(\rho)=2$.

\begin{prop}
     \label{ramseq2.prop}
     Let $\rho\in{}_{2}\RS$ be a $2$-torsion ramification sequence with
     $\deg\supp(\rho)=2$.
     \begin{enumerate}
	\item
	If $\supp(\rho)=\{p_{1},p_{2}\}$ (two distinct rational points), there
	is a nonsquare $a\in F^\times$ and a choice of projective
	coordinates $\PP^{1}_{F}=\Proj(F[u,v])$ such that
	\[
	p_{1}=(u-av)F[u,v],\qquad
	p_{2}=(u+av)F[u,v],
	\]
	and the nontrivial components of $\rho$ are
	\[
	aF^{\times2}\text{ at $p_{1}$},
	\qquad
	aF^{\times2}\text{ at $p_{2}$}.
	\]
	\item
	If $\supp(\rho)=\{q\}$ with $\deg q=2$, there are nonsquares $a$, $b\in
	F^\times$ in different square classes and a choice of projective
	coordinates $\PP^{1}_{F}=\Proj(F[u,v])$ such that
	\[
	q=(u^{2}-bv^{2})F[u,v]
	\]
	and the component of $\rho$ at $q$ is $aF^{\times2}_{q}$.
     \end{enumerate}
\end{prop}

\begin{proof}
     (1) The nontrivial components of $\rho$ are square classes in
     $F_{p_{1}}=F$ and $F_{p_{2}}=F$. These square classes coincide,
     since $\rho$ is in the kernel of the norm map \eqref{N.eq}. We may
     therefore find $a\in F^\times$ such that the nontrivial components
     of $\rho$ are $aF^{\times2}$ at $p_{1}$ and $p_{2}$. By
     Corollary~\ref{projco.cor}, we may choose projective coordinates
     such that $p_{1}=(u-av)F[u,v]$ and $p_{2}=(u+av)F[u,v]$.
     This completes the proof of~(1).
     \medbreak
     \par
     (2) We may find a nonsquare $b\in F^\times$ such that
     $F_{q}\simeq F(\sqrt{b})$, hence, by Corollary~\ref{projco.cor},
     there are projective coordinates such that $q=(u^{2}-bv^{2})F[u,v]$. The
     component of $\rho$ at $q$ is a square class in the kernel of the
     norm-induced map
     \[
     F_{q}^\times/F_{q}^{\times2}\to F^\times/F^{\times2}.
     \]
     As an application of Hilbert's Theorem~90, it is easily seen that
     the kernel of this map is the image of the map
     $F^\times/F^{\times2}\to F_{q}^\times/F_{q}^{\times2}$ induced by
     the inclusion $F\subset F_{q}$. (See \cite[p.~202]{Lam}.)
     Therefore, there is a
     nonsquare $a\in F^\times$ such that the component of $\rho$ at $q$
     is $aF_{q}^{\times2}$.
\end{proof}

We now turn to the case where the support of $\rho$ has degree~$3$.
Three cases arise:
\begin{enumerate}
     \item[(I)]
     $\supp(\rho)=\{p_{1},p_{2},p_{3}\}$ where $p_{1}$, $p_{2}$,
     $p_{3}$ are three distinct rational points;
     \item[(II)]
     $\supp(\rho)=\{p,q\}$ where $p$ is a rational point and $q$ is a
     point of degree~$2$;
     \item[(III)]
     $\supp(\rho)=\{r\}$, a single point of degree~$3$.
\end{enumerate}

In case (I), let $a_{i}F^{\times2}$ be the component of $\rho$ at $p_{i}$,
for $i=1$, $2$, $3$. Since these components are nontrivial,
$a_{i}\notin F^{\times2}$ for all $i$. However,
$a_{1}a_{2}a_{3}\in F^{\times2}$ since $\rho$ is in the kernel of the
norm map \eqref{N.eq}, hence we may substitute $a_{1}a_{2}$
for $a_{3}$. Therefore, discarding the trivial components of $\rho$,
we may write
\[
\rho=(a_{1}F_{p_{1}}^{\times2},a_{2}F_{p_{2}}^{\times2},
a_{1}a_{2}F_{p_{3}}^{\times2}).
\]
Applying Corollary~\ref{projco.cor}, we may choose projective
coordinates so that $\infty\neq p_{1}$, $p_{2}$, $p_{3}$, and in fact
we may assume
\[
p_{1} =(t-a_{1})F[t],\quad
p_{2} =(t-a_{2})F[t],\quad
p_{3} =(t-a_{1}a_{2})F[t].
\]

In case (II), let $y\in F_{q}^\times$ be such that the component of
$\rho$ at $q$ is $yF_{q}^{\times2}$.
Since $\rho$ is in the kernel of the norm
map \eqref{N.eq}, the component of $\rho$ at $p$ is
\[
N_{F_{q}/F}(y)F^{\times2}\in F^\times/F^{\times2}.
\]
Since this component is not trivial, we have $y\notin F$. Let
$X^{2}-mX+a$ be the minimal polynomial of $y$ over $F$, so
$a=N_{F_{q}/F}(y)$ and
\[
\rho=(aF_{p}^{\times2},yF_{q}^{\times2}).
\]
We may apply Corollary~\ref{projco.cor} to choose coordinates such
that
\[
p=(t-a)F[t],\qquad
q=(t^{2}-mt+a)F[t].
\]

In case (III), let $x\in F_{r}^\times$ be a nonsquare such that the
component of $\rho$ at $r$ is $xF_{r}^{\times2}$,
\[
\rho=(xF_{r}^{\times2}).
\]
Since $\rho$ is in the kernel of the norm map \eqref{N.eq}, we have
\[
N_{F_{r}/F}(x)=a^{2}\qquad\text{for some $a\in F^\times$}.
\]
If $x\in F$ or if $F_{r}/F$ is not separable, then $a^{2}=x^{3}$,
and we get a contradiction since $x\notin F_{r}^{\times2}$.
Let $X^{3}-cX^{2}+bX-a^{2}$ be the minimal polynomial of $x$ over
$F$. This is an irreducible separable polynomial in $F[X]$. By
Corollary~\ref{projco.cor}, we may assume
\[
r=(t^{3}-ct^{2}+bt-a^{2})F[t].
\]

We summarize the special choice of coordinates as follows:

\begin{prop}
     \label{ramseq3.prop}
     In all the cases (I), (II), (III), we may choose projective
     coordinates in
     $\PP^{1}_{F}$ so that $\infty\notin\supp(\rho)$ and
     for each $p\in\supp(\rho)$ the nontrivial component of $\rho$ at
     $p$ is $\tau F_{p}^{\times 2}$, where $\tau$ is the image of $t$
     under the canonical map $F[t]\to F[t]/p$.
\end{prop}

\begin{proof}
     Corollary~\ref{projco.cor} yields projective coordinates for which
     the ramification points have the required form. In case (II), the
     element $y\in F_{q}^\times$ is thus identified with one of the
     roots of $X^{2}-mX+a$ in $F[t]/q$. If it is not identified with
     $\tau$, Proposition~\ref{1+2.prop} still allows a change of
     coordinates which leaves $p$ and $q$ invariant and maps $y$ to
     $\tau$. Similarly, in case (III) we may assume $x\in
     F_{r}^\times$ is identified with $\tau$ by
     Proposition~\ref{3.prop}.
\end{proof}

\section{Algebras with $2$-torsion ramification sequence of degree 2}

In this section, the base field $F$ is assumed to be of
characteristic different from~$2$. Let $\rho\in{}_{2}\RS$ be a
$2$-torsion ramification sequence whose support has degree~$2$.
Proposition~\ref{ramseq2.prop} shows that, after a change of
projective coordinates, we may assume that either
\begin{enumerate}
     \item[(I)]
     there exists a nonsquare $a\in F^\times$ such that
     $\supp(\rho)=\{a,-a\}$ and the nontrivial components of $\rho$
     are $aF^{\times2}$, or
     \item[(II)]
     there exist nonsquares $a$, $b\in F^\times$ in different square
     classes such that $\supp(\rho)=\{q=(u^{2}-bv^{2})F[u,v]\}$ and
     the nontrivial component of $\rho$ is $aF_{q}^{\times2}$.
\end{enumerate}
In each case, it is not difficult to find a quaternion $F(t)$-algebra
$Q$ with ramification $\partial(Q)=\rho$, as we proceed to show.

Recall that for any closed point $p\in \PP^{1(1)}_{F}$, the image of a
quaternion algebra $(f,g)_{F(t)}$ under the ramification map
$\partial_{p}$ is given as follows:
\begin{equation}
     \label{ramap.eq}
\partial_{p}(f,g)_{F(t)}=(-1)^{v(f)v(g)}
\overline{f^{v(g)}g^{-v(f)}}F_{p}^{\times2}\in
F_{p}^{\times}/F_{p}^{\times2},
\end{equation}
where $v$ is the discrete valuation of $F(t)$ corresponding to $p$
and $\invo$ denotes the residue map from the valuation ring of $v$ to
its residue field $F_{p}$. This follows from the description of the
residue map (tame map) in Milnor's $K$-theory and the functoriality
of the norm residue homomorphism, see \cite[Theorem~2.3]{Bloch}.
Using this description, it is
easily verified that $\partial(a,t^{2}-a^{2})_{F(t)}$ is as in (I)
above, and that $\partial(a,t^{2}-b)_{F(t)}$ is as in (II). Therefore,
if $\alpha\in \Br F(t)$ has torsion prime to $\charac F$ and
a ramification sequence $\partial(\alpha)$
as in (I) (resp.\ (II)), then by Faddeev's exact
sequence \eqref{Fad.eq}, there exists a central simple $F$-algebra $A$
such that
\[
\alpha=[A\otimes_{F}(a,t^{2}-a^{2})_{F(t)}]
\qquad\bigl(\text{resp. }\alpha=[A\otimes_{F}(a,t^{2}-b)_{F(t)}]\bigr).
\]
We first compute the index $\ind\alpha$ in both cases simultaneously by
considering quaternion algebras of the type $(a,t^{2}-b)_{F(t)}$
where $a$, $b\in F^\times$ satisfy $a\notin F^{\times2}$ and
$ab\notin F^{\times2}$ (thus allowing $b=a^{2}$). The specific
features of each case will be discussed next.

\begin{thm}
     \label{locram2ind.thm}
     For every central simple $F$-algebra $A$ and $a$, $b\in F^\times$
     such that $a\notin F^{\times2}$ and $ab\notin F^{\times2}$,
     \[
     \ind(A\otimes_{F}(a,t^{2}-b)_{F(t)})=
     2\gcd\bigl\{
     \ind\bigl(A\otimes F(\sqrt{a})\bigr),
     \ind\bigl(A\otimes F(\sqrt{ab})\bigr)
     \bigr\}.
     \]
\end{thm}

Note that if $a\in F^{\times2}$ or $ab\in F^{\times2}$, then
$(a,t^{2}-b)_{F(t)}=(ab,t^{2}-b)_{F(t)}$ is split, hence obviously
\[
\ind(A\otimes_{F}(a,t^{2}-b)_{F(t)})=\ind A.
\]

\begin{proof}
     By general principles (see for instance \cite[p.~68]{Draxl}), for
     any quaternion $F(t)$-algebra $Q$ we have
     \[
     \ind(A\otimes_{F}Q)\text{ divides }2\ind(A)
     \]
     and, similarly, substituting $A\otimes_{F}Q$ for $A$,
     \[
     \ind(A)=\ind(A\otimes_{F}Q\otimes_{F(t)}Q)\text{ divides }
     2\ind(A\otimes_{F}Q).
     \]
     Therefore,
     \begin{equation}
	\label{tricho.eq}
	\ind(A\otimes_{F}Q)\in\bigl\{
	\half\ind A,\; \ind A,\; 2\ind A\bigr\}.
     \end{equation}
     Moreover, if $\ind(A\otimes_{F}Q)=\frac12\ind A$, then letting
     $D$ (resp.\ $D'$) be a division algebra Brauer-equivalent to $A$
     (resp.\ $A\otimes_{F}Q$), we have $\deg D(t)=\ind A$ and $\deg
     D'=\frac12\ind A$, hence
     \[
     \deg D(t)=\deg(D'\otimes_{F(t)}Q).
     \]
     Since $D(t)$ and $D'\otimes_{F(t)}Q$ are Brauer-equivalent, it
     follows that
     \[
     D(t)\simeq D'\otimes_{F(t)}Q,
     \]
     hence $D(t)$ contains a subalgebra isomorphic to $Q$.

     In the case where $Q=(a,t^{2}-b)_{F(t)}$, the last condition
     implies that $D(t)$ contains elements $f$, $g$ such that $fg=-gf$
     and $g^{2}=t^{2}-b$. Since $D(t)=D\otimes_{F}F(t)$, we may find
     $f_{0}$, $g_{0}\in F[t]$ and $f_{1}$, $g_{1}\in D[t]$ such that
     $f=f_{1}f_{0}^{-1}$ and $g=g_{1}g_{0}^{-1}$. Then, clearing
     denominators,
     \[
     f_{1}g_{1}=-g_{1}f_{1}
     \qquad\text{and}\qquad
     g_{1}^{2}=(t^{2}-b)g_{0}^{2}.
     \]
     \relax From these equations, it follows that the leading
     coefficient of $g_{1}$ is in $F$ and that it anticommutes with the
     leading coefficient of $f_{1}$, a contradiction. Therefore, the
     choices in \eqref{tricho.eq} restrict to
     \begin{equation}
	\label{dicho.eq}
	\ind(A\otimes_{F}(a,t^{2}-b)_{F(t)})\in
	\{\ind A,\;2\ind A\}.
     \end{equation}
     Clearly, $\ind(A\otimes_{F}(a,t^{2}-b)_{F(t)})=2\ind A$ holds if
     and only if $D\otimes_{F}(a,t^{2}-b)_{F(t)}$ is a division
     algebra, which occurs if and only if $D$ does not contain a copy
     of $F(\sqrt{a})$ nor of $F(\sqrt{ab})$, by
     Proposition~\ref{degg2.prop}. Now, by \cite[p.~67]{Draxl}, we have
     \[
     \ind\bigl(A\otimes F(\sqrt{a})\bigr)=
     \begin{cases}
	\ind A&\text{if $F(\sqrt{a})\not\hookrightarrow D$,}\\
	\frac12\ind A&\text{if $F(\sqrt{a})\hookrightarrow D$.}
     \end{cases}
     \]
     Similarly,
     \[
     \ind\bigl(A\otimes F(\sqrt{ab})\bigr)=
     \begin{cases}
	\ind A&\text{if $F(\sqrt{ab})\not\hookrightarrow D$,}\\
	\frac12\ind A&\text{if $F(\sqrt{ab})\hookrightarrow D$.}
     \end{cases}
     \]
     Summing up, we have $\ind(A\otimes(a,t^{2}-b)_{F(t)})=2\ind A$ if
     and only if
     \[
     \ind\bigl(A\otimes F(\sqrt{a})\bigr)=\ind A=
     \ind\bigl(A\otimes F(\sqrt{ab})\bigr).
     \]
     Therefore, by~\eqref{dicho.eq},
     $\ind(A\otimes(a,t^{2}-b)_{F(t)})=\ind A$ if and only if
     \[
     \ind\bigl(A\otimes F(\sqrt{a})\bigr)=\half\ind A
     \quad\text{or}\quad
     \ind\bigl(A\otimes F(\sqrt{ab})\bigr)=\half\ind A.
     \]
\end{proof}

We now consider separately the two cases for the support of the
ramification sequence.

\subsection{Two rational points}

Suppose $\rho\in{}_{2}\RS$ is a ramification sequence whose support
consists of two rational points. Let $aF^{\times2}$ be the component
of $\rho$ at these two points.

\begin{corol}
     \label{ramloc21.cor}
     The quaternion algebras $Q$ over $F(t)$ with $\partial(Q)=\rho$ form
     a $1$-parameter family. For any quaternion algebra $Q$ in this
     family and any central simple $F$-algebra $A$,
     \[
     \ind(A\otimes_{F}Q)=
     2\ind\bigl(A\otimes_{F}F(\sqrt{a})\bigr).
     \]
\end{corol}

\begin{proof}
     By Proposition~\ref{ramseq2.prop}, we may assume the two rational
     points in the support of $\rho$ are $a$ and $-a$, so that
     \[
     \partial(a,t^{2}-a^{2})_{F(t)}=\rho.
     \]
     Now, assume $Q$ is a quaternion $F(t)$-algebra such that
     $\partial(Q)=\rho$. By Faddeev's exact sequence~\eqref{Fad.eq},
     there is a central simple $F$-algebra $B$ such that
     \[
     [Q]=[B\otimes_{F}(a,t^{2}-a^{2})_{F(t)}]
     \qquad\text{in $\Br F(t)$}.
     \]
     Then $\ind(B\otimes_{F}(a,t^{2}-a^{2})_{F(t)})=2$, and
     Theorem~\ref{locram2ind.thm} implies that $B$ is split by
     $F(\sqrt{a})$. Therefore, $B$ is Brauer-equivalent to a
     quaternion algebra $(a,\lambda)_{F}$ for some $\lambda\in
     F^\times$, and
     \[
     Q\simeq\bigl(a,\lambda(t^{2}-a^{2})\bigr)_{F(t)}.
     \]
     For any central simple $F$-algebra $A$, the index of
     $A\otimes_{F}Q$ is computed by Theorem~\ref{locram2ind.thm}:
     \begin{multline*}
     \ind(A\otimes_{F}Q)=
     \ind(A\otimes_{F}(a,\lambda)_{F}\otimes_{F}(a,t^{2}-a^{2})_{F(t)})
     =\\
     2\ind\bigl(A\otimes_{F}(a,\lambda)_{F}\otimes F(\sqrt{a})\bigr).
     \end{multline*}
\end{proof}

\begin{remark}
     If the two rational points in the support of $\rho$ are chosen to be
     $0$ and $\infty$, the family of quaternion algebras with
     $\partial(Q)=\rho$ is
     \[
     \{(a,\lambda t)_{F(t)}\mid\lambda\in F^\times\}.
     \]
     The formula $\ind(A\otimes_{F}(a,\lambda t)_{F(t)})=2
     \ind\bigl(A\otimes F(\sqrt{a})\bigr)$ was obtained by a different
     proof in \cite[Proposition~2.4]{Tig}.
\end{remark}

\subsection{Single point of degree 2}

Suppose $\rho\in{}_{2}\RS$ is a ramification sequence whose support
consists of a single point $q$ of degree~$2$.

\begin{corol}
     \label{ramloc22.cor}
     The quaternion $F(t)$-algebras $Q$ with $\partial(Q)=\rho$ form two
     $1$-parameter families.
\end{corol}

\begin{proof}
     As observed in Proposition~\ref{ramseq2.prop}, we may find
     nonsquares $a$, $b\in F^\times$ in different square classes such
     that
     $q=(u^{2}-bv^{2})F[u,v]$, hence $F_{q}=F(\sqrt{b})$, and the
     nontrivial component of $\rho$ is $aF_{q}^{\times2}$.
     Then,
     \[
     \partial(a,t^{2}-b)_{F(t)}=\rho.
     \]
     If $Q$ is a quaternion $F(t)$-algebra with $\partial(Q)=\rho$, then
     by Faddeev's exact sequence~\eqref{Fad.eq} there exists a central
     simple $F$-algebra $B$ such that
     \[
     [Q]=[B\otimes_{F}(a,t^{2}-b)_{F(t)}]
     \qquad\text{in $\Br F(t)$}.
     \]
     By Theorem~\ref{locram2ind.thm}, the equation
     $\ind(B\otimes_{F}(a,t^{2}-b)_{F(t)})=2$ implies that $B$ is split
     by $F(\sqrt{a})$ or by $F(\sqrt{ab})$, hence $B$ is
     Brauer-equivalent to $(a,\lambda)_{F}$ or to $(ab,\lambda)_{F}$
     for some $\lambda\in F^\times$. Therefore, we have either
     \[
     Q\simeq \bigl(a,\lambda(t^{2}-b)\bigr)_{F(t)}
     \qquad\text{or}\qquad
     Q\simeq\bigl(ab,\lambda(t^{2}-b)\bigr)_{F(t)}.
     \]
\end{proof}

\begin{remark}
     Of course, the two families in the corollary above are not
     disjoint. For $\lambda\in F^\times$ there exists $\mu\in F^\times$
     such that
     \[
     \bigl(ab,\lambda(t^{2}-b)\bigr)_{F(t)}=
     \bigl(a,\mu(t^{2}-b)\bigr)_{F(t)}
     \]
     if and only if $(b,\lambda)_{F}$ is split by $F(\sqrt{a})$. The
     easy proof is omitted.
\end{remark}

\begin{remark}
     As in Corollary~\ref{ramloc21.cor}, we may use
     Theorem~\ref{locram2ind.thm} to determine the index of
     $A\otimes_{F}\bigl(a,\lambda(t^{2}-b)\bigr)_{F(t)}$ for any
     $\lambda\in F^\times$, but the result depends on the parameter
     $\lambda$:
     \begin{multline*}
     \ind\bigl(A\otimes_{F}\bigl(a,\lambda(t^{2}-b)\bigr)_{F(t)}\bigr)
     =\\
     2\gcd\bigl\{
     \ind\bigl(A\otimes F(\sqrt{a})\bigr),\;
     \ind\bigl(A\otimes(a,\lambda)_{F}\otimes F(\sqrt{ab})\bigr)
     \bigr\}.
     \end{multline*}
     Similarly,
     \begin{multline*}
     \ind\bigl(A\otimes_{F}\bigl(ab,\lambda(t^{2}-b)\bigr)_{F(t)}\bigr)
     =\\
     2\gcd\bigl\{
     \ind\bigl(A\otimes(ab,\lambda)_{F}\otimes F(\sqrt{a})\bigr),\;
     \ind\bigl(A\otimes F(\sqrt{ab})\bigr)
     \bigr\}.
     \end{multline*}
\end{remark}

\section{Algebras with $2$-torsion ramification sequence of degree 3}
\label{ramloc3.sec}

As in the preceding section, the base field $F$ is assumed to be of
characteristic different from~$2$. Let $\rho\in{}_{2}\RS$ be a
$2$-torsion ramification sequence with support of degree~$3$.
We use the same notation as in Section~\ref{ramseq.sec}, and assume
projective coordinates have been chosen as in
Proposition~\ref{ramseq3.prop}. For each $p\in\supp(\rho)$, let
$P_{\rho}\in F[t]$ be the monic irreducible polynomial such that
$p=P_{p}F[t]$. Let also
\[
P_{\rho}=\prod_{p\in\supp(\rho)}P_{p},
\qquad
F_{\rho}=F[t]/P_{\rho}F[t]=\prod_{p\in\supp(\rho)}F_{p},
\]
and let $x_{\rho}\in F_{\rho}$ be the image of $t$ in $F_{\rho}$. By
Proposition~\ref{ramseq3.prop}, we have
$\rho=x_{\rho}F_{\rho}^{\times2}$. Moreover, as observed in
Section~\ref{ramseq.sec}, $F_{\rho}$ is an \'etale $F$-algebra.

With the notation and choice of coordinates above, it is readily
verified that the quaternion algebra
\begin{equation}
     \label{Q.eq}
Q=(t,-P_{\rho})_{F(t)}
\end{equation}
satisfies $\partial Q=\rho$ in each case (I), (II), and (III).
Therefore, if $\alpha\in \Br F(t)$ has torsion prime to $\charac F$ and
satisfies $\partial\alpha=\rho$,
then Faddeev's exact sequence \eqref{Fad.eq} yields a central simple
$F$-algebra $A$ such that
\[
\alpha=[A\otimes_{F}(t,-P_{\rho})_{F(t)}].
\]

In order to determine the index $\ind\alpha$, we use a quartic
\'etale $F$-algebra $E_{\rho}$ canonically associated with $\rho$,
viewed as the quadratic extension $F_{\rho}(\sqrt{x_{\rho}})/F_{\rho}$.
The construction of a quartic \'etale $F$-algebra from a quadratic
\'etale extension of a cubic \'etale $F$-algebra is discussed at
length in \cite[\S5]{KT}. We briefly recall the cohomological version of
this construction.

For any integer $n$, let $\Sym_{n}$ be the symmetric group on $n$
elements. The conjugation action of the wreath product
$\Sym_{3}\wr\Sym_{2}$ on its four Sylow $3$-subgroups yields a map
\[
s\colon\,\Sym_{3}\wr\Sym_{2}\to\Sym_{4}.
\]
Let $\Gamma$ be the absolute Galois group of $F$. The Galois
cohomology set $H^{1}(\Gamma,\Sym_{3}\wr\Sym_{2})$ (for
the trivial $\Gamma$-action) classifies the isomorphism classes of
quadratic extensions of cubic \'etale $F$-algebras, so
$\rho$ defines an element in this set (corresponding to the
extension $F_{\rho}(\sqrt{x_{\rho}})/F_{\rho}$). Consider the map
induced by $s$,
\[
s_{*}\colon\,H^{1}(\Gamma,\Sym_{3}\wr\Sym_{2})\to
H^{1}(\Gamma,\Sym_{4}).
\]
Since $H^{1}(\Gamma,\Sym_{4})$ classifies quartic \'etale
$F$-algebras up to isomorphism, the image of $\rho$
under this map defines a quartic \'etale $F$-algebra $E_{\rho}$.

An explicit description of $E_{\rho}$ is given in \cite[\S5.4]{KT}. Note that
in each case (I, II or III), $x_{\rho}$ generates $F_{\rho}$ as an
$F$-algebra, with minimal polynomial $P_{\rho}$ of the form
\[
P_{\rho}=t^{3}-ct^{2}+bt-a^{2}\in F[t].
\]
(The norm $N_{F_{\rho}/F}(x_{\rho})$ is a square because $\rho$ lies
in the kernel of the sum of corestriction maps
$\bigoplus_{p}X(F_{p})\to X(F)$.)
Applying \cite[Corollary~5.22]{KT}, we may describe $E_{\rho}$ as the
quartic $F$-algebra generated by an element with minimal polynomial
\begin{equation}
     \label{quminpol.eq}
X^{4}-2cX^{2}-8aX+(c^{2}-4b).
\end{equation}

Remarkably, the classification in \cite[\S6.3]{KT} shows that the
$F$-algebra $E_{\rho}$ is a field in each case (I, II, or III). In
case (I), we have
\[
P_{\rho}=(t-a_{1})(t-a_{2})(t-a_{1}a_{2}),\qquad
E_{\rho}\simeq F(\sqrt{a_{1}},\sqrt{a_{2}}),
\]
and \eqref{quminpol.eq} is the minimal polynomial of
$\sqrt{a_{1}}+\sqrt{a_{2}}+\sqrt{a_{1}a_{2}}$. In case (II), $E_{\rho}/F$
is a quartic $2$-extension, with intermediate quadratic extension
$F(\sqrt{a})$. In case (III), $E_{\rho}/F$ is a
quartic extension whose Galois closure has Galois group $\Sym_{4}$ or
the alternating group $\mathfrak{A}_{4}$.

\begin{remark}
     The table in \cite[\S6.3]{KT} is set up from the perspective of
     the quartic algebra. The correspondence with the notation used
     here is the following: $Q=E_{\rho}$, $\mathcal{R}(Q)=F_{\rho}$,
     $\Lambda_{2}(Q)=F_{\rho}(\sqrt{x_{\rho}})$.
\end{remark}

\begin{thm}
     \label{locram3ind.thm}
     For every central simple $F$-algebra $A$ and $P_{\rho}\in F[t]$ as
     above,
     \[
     \ind(A\otimes_{F}
     (t,-P_{\rho})_{F(t)})=
     \begin{cases}
	\ind A&\text{if $\ind(A\otimes_{F}E_{\rho})=
	\frac14\ind A$},\\
	2\ind A&\text{otherwise}.
     \end{cases}
     \]
\end{thm}

\begin{proof}
     Let $D$ be a central division $F$-algebra Brauer-equivalent to $A$.
     As observed in the beginning of the proof of
     Theorem~\ref{locram2ind.thm} (see \eqref{tricho.eq}),
     \[
     \ind(A\otimes_{F}
     (t,-P_{\rho})_{F(t)})=\half\ind A,\;
     \ind A\text{ or }2\ind A.
     \]
     Moreover, the first case occurs only if $D(t)$ contains a
     subalgebra isomorphic to
     $(t,-P_{\rho})_{F(t)}$. In particular, it must
     contain a square root of $t$, hence $D(t)$ does not remain a
     division algebra over $F(t)(\sqrt{t})$. This is impossible, since
     $F(t)(\sqrt{t})$ is a purely transcendental extension of $F$.
     Therefore, only two
     possibilities remain:
      \[
     \ind(A\otimes_{F}
     (t,-P_{\rho})_{F(t)})=
     \ind A\text{ or }2\ind A.
     \]
     The first case occurs if and only if
     $D\otimes(t,-P_{\rho})_{F(t)}$ is not a division
     algebra. By Proposition~\ref{degg3.prop}, this condition is
     equivalent to the existence of a subalgebra isomorphic to
     $E_{\rho}$ in $D$, hence also, by \cite[Theorem~12, p.~67]{Draxl}, to
     \[
     \ind(D\otimes_{F}E_{\rho})={\textstyle\frac14}\ind D.
     \]
\end{proof}

\begin{corol}
     \label{ramloc3.cor}
     For any ramification sequence $\rho\in{}_{2}\RS$ whose support
     has degree~$3$,
     there is (up to isomorphism) a unique quaternion
     $F(t)$-algebra $Q_{\rho}$ such that
     \[
     \partial Q_{\rho}=\rho.
     \]
\end{corol}

\begin{proof}
     By a suitable choice of projective coordinates, we may assume
     $\rho$ has the special form of Proposition~\ref{ramseq3.prop}. As
     observed above (see \eqref{Q.eq}), the quaternion algebra
     \[
     Q=(t,-P_{\rho})_{F(t)}
     \]
     satisfies $\partial Q=\rho$. If $Q'$ is a quaternion
     $F(t)$-algebra such that $\partial Q'=\rho$, then by Faddeev's
     exact sequence \eqref{Fad.eq}, there is a central division
     $F$-algebra $D$ such that $Q'$ is Brauer-equivalent to
     $D\otimes_{F}Q$. By Theorem~\ref{locram3ind.thm}, the equality
     $\ind(D\otimes_{F}Q)=2$ leads to $\ind D=1$ (hence $Q'\simeq
     Q$), or $\ind D=2$. The latter case occurs only if
     \[
     \ind(D\otimes E_{\rho})={\textstyle\frac14}\ind D=\half,
     \]
     which is impossible.
\end{proof}

Thus, if $\rho$ is in the special form of
Proposition~\ref{ramseq3.prop},
\[
Q_{\rho}=(t,-P_{\rho})_{F(t)}.
\]
The index of any Brauer class $\alpha$ such that $\partial\alpha=\rho$
is determined by Theorem~\ref{locram3ind.thm}.

\begin{corol}
     \label{ex.cor}
     If $a$, $b\in F^\times$ are nonsquares in different square classes
     and $D$ is a quaternion division $F$-algebra, the tensor product
     \[
     D\otimes_{F}\bigl(t,a(b-t)\bigr)_{F(t)}
     \]
     is a division algebra.
\end{corol}

\begin{proof}
     Inspection shows that $\bigl(t,a(b-t)\bigr)_{F(t)}$ ramifies only
     at $0$, $b$, and $\infty$, with respective ramification
     $abF^{\times2}$, $bF^{\times2}$, and $aF^{\times2}$. Therefore,
     for a suitable change of variables,
     \[
     \bigl(t,a(b-t)\bigr)_{F(t)}=
     \bigl(t',(a-t')(b-t')(ab-t')\bigr)_{F(t')}.
     \]
     Theorem~\ref{locram3ind.thm} shows that
     $D\otimes_{F}\bigl(t,a(b-t)\bigr)_{F(t)}$ is not a division
     algebra if and only if
     \[
     \ind\bigl(D\otimes_{F}F(\sqrt{a},\sqrt{b})\bigr)=
     {\textstyle\frac14}\ind D=\half,
     \]
     which is absurd.
\end{proof}

As a particular case, one may take $F=\QQ_{p}$ (the field of $p$-adic
numbers) with $p\neq2$, $a\in\ZZ_{p}$ a nonsquare unit, $b=p$ and
$D=(a,p)_{\QQ_{p}}$. Corollary~\ref{ex.cor} shows that the tensor
product
\[
(a,p)_{\QQ_{p}}\otimes_{\QQ_{p}}\bigl(t,a(p-t)\bigr)_{\QQ_{p}(t)}
\]
is a division algebra over $\QQ_{p}(t)$. Note that by a theorem of
Saltman~\cite[Theorem~3.4]{Salt}, division algebras of exponent~$2$ over
$\QQ_{p}(t)$ have index~$2$ or $4$. The Jacob--Tignol example of a
biquaternion division algebra over $\QQ_{p}(t)$ given in the appendix
of \cite{Salt} is not essentially different from the example above,
but the arguments used to prove it to be a division algebra are quite
different.

\section*{Appendix: Tensor product with a symbol of degree 3}

Theorem~\ref{Delta.thm} has been used so far to investigate tensor
products with quaternion algebras only. In this appendix, we give
another application, where the prime $n$ in the statement of
Theorem~\ref{Delta.thm} is $3$ instead of $2$. This
will illustrate the fact that Theorem~\ref{Delta.thm} leads to more
complex conditions for $n>2$.

Let $F$ be a field of characteristic different from~$3$, containing a
primitive cube root of unity $\zeta$. For $a$, $b\in F^\times$, we
denote by $(a,b)_{\zeta,F}$ the \emph{symbol algebra} of degree~$3$
generated by two elements $i$, $j$ subject to
\[
i^{3}=a,\qquad ji=\zeta ij,\qquad j^{3}=b.
\]
(See \cite[\S11]{Draxl}, where this type of algebra is called
\emph{power norm residue algebra}.)

\begin{prop}
     \label{app.prop}
     Let $D$ be a central division $F$-algebra, let $a$, $b\in
     F^\times$ and let $t$ be an indeterminate over $F$. The tensor
     product
     \begin{equation}
	\label{TP.eq}
	D\otimes_{F}(t,at+b)_{\zeta,F(t)}
     \end{equation}
     is not a division algebra if and only if $D$ contains elements
     $u$, $v$ satisfying
     \begin{equation}
	\label{cond3.eq}
	\begin{split}
	    u^{3}=a,\qquad&\qquad
	    u(uv-vu)=\zeta(uv-vu)u,\\
	    v^{3}=b,\qquad&\qquad
	    v(uv-vu)=\zeta^{2}(uv-vu)v.
	\end{split}
     \end{equation}
\end{prop}

\begin{proof}
     The tensor product \eqref{TP.eq} can be viewed as a special case
     of the $\Delta$-construction in \eqref{Delta.eq},
     \[
     D\otimes_{F}(t,at+b)_{\zeta,F(t)}=
     \Delta(D(x),\alpha,ax^{3}+b)
     \]
     where $x^{3}=t$ and $\alpha$ is the automorphism of $D(x)$
     defined by
     \[
     \alpha(x)=\zeta x,\qquad
     \alpha(d)=d\quad\text{for $d\in D$}.
     \]
     (Compare Section~\ref{indetslot.subsec}.) By
     Theorem~\ref{Delta.thm}, this algebra is not a division algebra
     if and only if there exists $f\in D[x]$ such that
     \begin{equation}
	\label{f.eq}
	\alpha^{2}(f)\alpha(f)f=ax^{3}+b.
     \end{equation}
     Comparing degrees, it is clear that $f$ must be of the form
     $f=ux+v$ for some $u$, $v\in D$ if it exists. Comparing
     coefficients of like powers of $x$ in \eqref{f.eq} (and using
     $\zeta^{2}=-1-\zeta$) yields
     \eqref{cond3.eq}.
\end{proof}

For instance, the tensor product \eqref{TP.eq} is not a division
algebra if $D\supset F(\sqrt[3]{a},\sqrt[3]{b})$, since then
\eqref{cond3.eq} has a solution where $u$ and $v$ commute. This is not
the only case, however:

\begin{corol}
     \label{app1.cor}
     The tensor product
     \[
     (b,a)_{\zeta,F}\otimes_{F}(t,at+b)_{\zeta,F(t)}
     \]
     is not a division algebra.
\end{corol}

\begin{proof}
     The elements $u$, $v\in(b,a)_{\zeta,F}$ such that
     \[
     v^{3}=b,\qquad uv=\zeta vu,\qquad u^{3}=a
     \]
     satisfy \eqref{cond3.eq}.
\end{proof}

A sharp contrast is given by the following positive result:

\begin{corol}
     \label{app2.cor}
     Suppose $a$, $b$ are indeterminates over a field $k$ containing a
     primitive cube root of unity, and let $F=k(a,b)$. The tensor
     product
     \[
     (a,b)_{\zeta,F}\otimes_{F}(t,at+b)_{\zeta,F(t)}
     \]
     is a division algebra.
\end{corol}

\begin{proof}
     Embedding $F$ into the field of iterated Laurent series $\hat
     F=k((a))((b))$, we show the stronger result that
     \[
     (a,b)_{\zeta,\hat F}\otimes_{\hat F}(t,at+b)_{\zeta,\hat F(t)}
     \]
     is a division algebra.

     Observe that the canonical $(a,b)$-adic valuation on $\hat F$
     (with values in $\ZZ\times\ZZ$) extends to a totally ramified
     valuation $\nu$ on $(a,b)_{\zeta,\hat F}$. The standard
     generators $i$, $j$ of $(a,b)_{\zeta,\hat F}$ satisfy
     \[
     \nu(i)=\bigl({\textstyle\frac13},0\bigr),
     \qquad
     \nu(j)=\bigl(0,{\textstyle\frac13}\bigr).
     \]
     Suppose $u$, $v\in(a,b)_{\zeta,\hat F}$ satisfy \eqref{cond3.eq}.
     Then
     \[
     \nu(u)=\bigl({\textstyle\frac13},0\bigr)=\nu(i),
     \qquad
     \nu(v)=\bigl(0,{\textstyle\frac13}\bigr)=\nu(j),
     \]
     hence, by \cite[\S3]{TW}, letting $\invo$ denote the residue map,
     \[
     \overline{v^{-1}u^{-1}vu}=
     \overline{j^{-1}i^{-1}ji}=\zeta
     \]
     and
     \[
     \overline{v^{-1}u^{-2}vu^{2}}=
     \overline{j^{-1}i^{-2}ji^{2}}=\zeta^{2}.
     \]
     On the other hand, from the equation
     \[
     u(uv-vu)=\zeta(uv-vu)u
     \]
     it follows that
     \[
     u^{2}v+\zeta^{2}uvu+\zeta vu^{2}=0
     \]
     hence, dividing by $u^{2}v$,
     \[
     1+\zeta^{2}v^{-1}u^{-1}vu+\zeta v^{-1}u^{-2}vu^{2}=0.
     \]
     Taking the residue of each side yields $3=0$.
\end{proof}

\begin{remark}
     Letting $K=k(a,b,t)=F(t)$ and
     \[
     D_{1}=(a,b)_{\zeta,K},\qquad
     D_{2}=(t,at+b)_{\zeta,K},
     \]
     Corollaries~\ref{app1.cor} and \ref{app2.cor} show that
     $D_{1}\otimes_{K}D_{2}$ is a division algebra while
     $D_{1}^{\text{op}}\otimes_{K}D_{2}$ is not a division algebra.
     The first examples
     of this type are in \cite[\S5]{TW}. The example above is
     essentially the same as the Tignol--Wadsworth example in degree~$3$.
\end{remark}

\begin{remark}
     Suppose $D$ is a central division $F$-algebra of degree~$3$. If
     $u$, $v\in D$ satisfy \eqref{cond3.eq} and $uv-vu\neq0$, then
     \[
     (v,uv-vu,u)
     \]
     is a chain of length~$2$ of Kummer elements, in the sense of
     Rost \cite{Rost}. The example in \cite[Appendix]{Rost} of a
     division algebra containing two Kummer elements which cannot be
     related by a chain of length~$2$ yields, by
     Proposition~\ref{app.prop}, a division algebra of exponent~$3$ and
     degree~$9$ over the field of rational fractions in two
     indeterminates $\FF_{7}(t_{1},t_{2})$.
\end{remark}

\begin{thebibliography}{xx}

\bibitem{SofA}
A.A.~Albert, \textit{Structure of algebras}, Amer. Math. Soc.,
Providence, R.I., 1961. MR0123587 (23 \#A912)

\bibitem{Baer}
R.~Baer, \textit{Linear algebra and projective geometry}, Academic
Press, New York, N. Y., 1952. MR0052795 (14,675j)

\bibitem{Bloch}
S.~Bloch, \textit{Torsion algebraic cycles, $K_{2}$, and Brauer
groups of function fields}, in {\it The Brauer group (Sem.,
Les Plans-sur-Bex,
1980)}, 75--102, Lecture Notes in Math., 844, Springer, Berlin.
MR0611866 (82k:14018)

\bibitem{Draxl}
P.K.~Draxl, \textit{Skew fields}, Cambridge Univ. Press,
Cambridge, 1983. MR0696937 (85a:16022)

\bibitem{Fad}
D.K.~Faddeev, \textit{Simple algebras over a field of algebraic functions
of one variable},
in Trudy Mat. Inst. Steklov., v. 38, 321--344,
Akad. Nauk SSSR, Moscow. MR0047632 (13,905c);
Amer. Math. Soc. Transl. (2) \textbf{3} (1956), 15--38. MR0077505 (17,1046e)

\bibitem{GMS}
S.~Garibaldi, A.~Merkurjev\ and\ J.-P.~Serre, \textit{Cohomological
invariants in Galois cohomology}, Amer. Math. Soc., Providence,
RI, 2003. MR1999383

\bibitem{Jake}
N.~Jacobson, \textit{Finite-dimensional division algebras over fields},
Springer, Berlin, 1996. MR1439248 (98a:16024)

\bibitem{KT}
M.-A.~Knus, J.-P.~Tignol, \textit{Quartic Exercises},
IJMMS \textbf{2003} (2003), no.~68, 4263--4323.

\bibitem{KRTY}
B.\`E.~Kunyavski\u\i, L.H.~Rowen, S.V.~Tikhonov, V.I.~Yanchevki\u\i,
\textit{Bicyclic algebras of prime exponent over function fields}, to
appear.

\bibitem{Lam}
T.Y.~Lam, \textit{The algebraic theory of quadratic forms}, W. A.
Benjamin, Inc., Reading, Mass., 1973. MR0396410 (53 \#277)

\bibitem{Tig}
J.-P.~Tignol, \textit{Alg\`ebres ind\'ecomposables d'exposant premier},
Adv. in Math. \textbf{65} (1987), no.~3, 205--228. MR0904723 (88h:16028)

\bibitem{TW}
J.-P.~Tignol\ and\ A.R.~Wadsworth, \textit{Totally ramified valuations on
finite-dimensional division algebras}, Trans. Amer. Math. Soc.
\textbf{302} (1987), no.~1, 223--250. MR0887507 (88j:16025)

\bibitem{Rost}
M.~Rost, \textit{The chain lemma for Kummer elements of degree 3},
C. R. Acad. Sci.
Paris S\'er. I Math. \textbf{328} (1999), no.~3, 185--190. MR1674602
(2000c:12003)

\bibitem{Salt}
D.J.~Saltman, \textit{Division algebras over $p$-adic curves},
J. Ramanujan Math. Soc. \textbf{12} (1997), no.~1, 25--47. MR1462850
(98d:16032) \textit{Correction:} J. Ramanujan Math. Soc. 13 (1998),
no. 2, 125--129. MR1666362 (99k:16036)

\end{thebibliography}
\end{document}
