

%Cyclicity of algebras after a scalar extension
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%\topmatter
\begin{document}
\title[Symbols and cyclicity of algebras]
{Symbols and cyclicity of algebras after a scalar extension}
\author[Rehmann, Tikhonov, Yanchevski\u\i ]
{Ulf Rehmann, Sergey V. Tikhonov, and Vyacheslav I. Yanchevski\u\i
}
\thanks{The second
and the third authors are grateful to the CRC 701 at the University of Bielefeld
for the hospitality during the preparation of the paper}

\date{\today }
\address{Rehmann:
Universit\"at Bielefeld, POB 10 01 31, 33501 Bielefeld, GERMANY}
\email{rehmann@@math.uni-bielefeld.de}
\address{Tikhonov, Yanchevski\u\i : Institute of Mathematics
of the National Academy of Sciences of Belarus, ul. Surganova 11,
220072 Minsk, BELARUS} \email{tsv@@im.bas-net.by ,
yanch@@im.bas-net.by}

%Script
\def\cA{{\cal A}}
\def\cB{{\cal B}}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\cV{{\cal V}}
\def\cE{{\cal E}}
\def\cM{{\cal M}}
\def\cS{{\cal S}}
\def\Symb{\textrm{Symb}}
\def\Gal{\textrm{Gal}}

\begin{abstract}
%We examine cyclicity of algebras after scalar extensions.
1. For a field $F$ and a family of central simple $F$-algebras we
prove that there exists a regular field extension $E/F$ preserving
indices of $F$-algebras such that all the algebras from the family
are cyclic after scalar extension by $E$.

2. Let $\cA$ be a central simple algebra over a field $F$ of
degree $n$ with a primitive $n$-th root of unity $\rho_n$. We
construct a quasi-affine $F$-variety $\Symb(\cA)$ such that, for a
field extension $L/F$, the variety  $\Symb(\cA)$ has an $L$-rational point iff
$\cA \otimes_F L$ is a symbol algebra.

3. Let $\cA$ be a central simple algebra over a field $F$ of
degree $n$ and $K/F$ a cyclic field extension of degree $n$.  We
construct a quasi-affine $F$-variety $C(\cA, K)$ such that, for a
field extension $L/F$ with the property $[K L:L]=[K:F]$ ,the variety $C(\cA,
K)$ has an $L$-rational point iff $K L$ is a subfield of $\cA \otimes_F L$.
\end{abstract}

\maketitle

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Let $\cA$ be a finite dimensional central simple algebra over a field $F$. By Wedderburn's theorem, there is a 
unique integer $m\ge 1$ and a central division $F$-algebra $\cD$ which is unique up to $F$-isomorphism such that
$\cA\cong M_m(\cD)$. The degree of $\cA$ is defined by $\deg(\cA)=\sqrt{\dim_F \cA}$, the index of $\cA$ is said to be
$\ind(\cA)=\deg(\cD)$, and the exponent $\exp(\cA)$ of $\cA$ is the order of the equivalence class $[\cA]$ in the Brauer group
$\Br(F)$.  
%let $F$ be a field, $\Br(F)$ its Brauer group. The
%index and exponent of a central simple algebra $\cA$ are denoted
%by $\ind(\cA)$ and $\exp(\cA)$ respectively. 
For a field
extension $E/F$, $\res: \Br(F) \longrightarrow \Br(E)$ denotes the
restriction homomorphism and $\cA_E$ denotes the tensor product $\cA\otimes_F E$.


In the paper we consider the following problems related to properties of central 
simple algebras over scalar extensions.

\begin{problem} \label{prob1}
Let $\{\cA_{\alpha}\}_{\alpha \in I}$ be some family
of central simple $F$-algebras.  Fix some central simple algebra property $\mathcal{P}$.
Does there exist a field extension
$E/F$ (if possible, regular) such that each member of the family $\{
\cA_{\alpha}\otimes_F E\}_{\alpha \in I}$ has the property $\mathcal{P}$?
\end{problem}

\begin{problem} \label{prob2}
Let $\cA/F$ be a central simple algebra. Fix some central simple algebra property $\mathcal{P}$.
Does there exists an algebraic $F$-variety $V$ (affine,
quasi-affine, projective or quasi-projective) such that $\cA\otimes_F
L$ has the property $\mathcal{P}$ iff $V$ has an $L$-rational point?
\end{problem}


 Of course, the above problems in their all generality
are vague, so below we explain (and motivate) which kind of
properties $\mathcal{P}$ we mean.

%First we recall some classical notions and definitions related to
%central simple algebras $A/F$.
%
%\begin{definition}  The \textit{dimension} of $A$ ($\dim(A)$)
%is the dimension of the $F$-vector space $A$.
%\end{definition}
%
%\begin{definition} Let $\dim(A)=n^2$. The \textit{degree} of $A$ ($\deg(A)$) is number $n$.
%\end{definition}
%
%\begin{definition} If $A\cong M_n(D)$, where $D$ is a
%division $F$-algebra, then the \textit{index} of $A$ ($\ind(A)$) is
%$\deg(D)$.
%\end{definition}
%
%\begin{definition} Let $\Br(F)$ be the Brauer group of $F$,
%then the order of an element $[A]\in  \Br(F)$ with a representative
%$A$ is called the exponent of $A$ ($exp(A)$) (or period of $A$).
%\end{definition}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{example}
$\mathcal{P}$ is a property to have a given
index.
\end{example}

\begin{example} $\mathcal{P}$ is a property to have a given
exponent.
\end{example}

\begin{example}$\mathcal{P}$ is a property to have given
both exponent and index.
\end{example}


In \cite[Lemma 1.3.]{ReKe98} it is proved that given a central simple algebra
$\cA$ over $F$ of exponent $e$ and index $d$ with prime power decomposition $d=\prod p^{v_p(d)}$, 
then for any divisor $\delta$ of $d$ there exists a field extension $E^{(\delta)}$ such that
$$
\exp(\cA \otimes_F E^{(\delta)}) = \gcd (e,\delta) \; \mbox{ and } \ind(\cA \otimes_F E^{(\delta)}) = \prod_{p| \delta} p^{v_p(d)} .
$$ 

\begin{example} $\mathcal{P}$ is a property for an algebra to
have coincided exponent and index (index-exponent property).
\end{example}

\begin{remark}
 The latter Example leads us to the index-exponent problem.
\begin{problem}
Describe a class of fields $F$ for which $\ind(\cA)=\exp(\cA)$ for any
central simple $F$-algebra.
\end{problem}
It is an old open question whether this problem has a positive solution
in the class of $C_2$-fields. 
The recent results of A.~J.~de~Jong (\cite{Jong}) and M.~Lieblich 
(\cite{Li08}) show that for the class of function fields of surfaces over
algebraically closed fields this question has an affirmative answer.
\end{remark}

%%%%%%%%%%%%%%%%%%


Another kind of properties $\mathcal{P}$ is related to presentation
of algebras by generators and defining relations. The most popular
and simple classes 
of algebras in this sense are the following.

(i) \textit{Matrix algebras} $M_n(F)$. 
%Such algebras are the vector
%$F$-spaces with an $F$-base $\{e_{ij}\}_{i,j=1,\dots n}$ and the
%multiplication table:
%$$
%e_{ij}e_{kl} = \left\{
%\begin{array}{cc}
%e_{il},& \hbox{ if } j=k;\\
%0,& \hbox{ if } j\ne k;,
%\end{array}
%\right.
%$$

(ii) \textit{Crossed products} $(L/F,\Gal(L/F),f)$. Let $L/F$ be a
Galois field extension, $\Gal(L/F)$ its Galois group and $f$ a
2-cocycle of $G$ with values in $L^*$.
Then $(L/F,\Gal(L/F),f)$ is a left $L$-module with $L$-base
$\{u_{\tau}\}_{\tau \in \Gal(L/F)}$ and multiplication table:
$$
u_s l = l^s u_s, \; u_su_t=f(s,t)u_{st}
$$
for any $s,t \in \Gal(L/F)$ and $l\in L$.

(iii) \textit{Cyclic algebras} $(Z/F,s,a)$. They are a special form of
crossed products. Let $Z/F$ be a cyclic field extension of degree
$n$, $s$ a generator of $\Gal(Z/F)$ and $a\in F^*$. Then
$(Z/F,s,a)$ is a left $Z$-module with $Z$-base
$\{u_s^i\}_{i=1,\dots,n}$ and multiplication table:
$$
u^i_s c = c^{s^i} u^i_s
$$
and
$$
u^n_s = a
$$
for any $i=1,\dots,n$ and $c\in Z$.

(iv) \textit{Symbol algebras} $(a,b)_n$. These
algebras also have a simple set of generators and defining
relations. Let $\rho_n \in F$ be a primitive root of unity of
degree $n$ and $a,b \in F^*$. Then $(a,b)_n$ is an
$n^2$-dimensional vector $F$-space with an $F$-base
$$
\{A^i B^j\}_{i,j=1,\dots,n}
$$
and multiplication table
$$
A^j B^i=\rho_n ^j  B^i A^j, \; A^n =a, B^n =b.
$$

\begin{remark}	 Because of the simplicity of sets of the above generators and
defining relations it is interesting to study the properties
$\mathcal{P}$  of algebras to be one of the kinds above four types
(i)-(iv), especially to be cyclic or symbol (more generally to be a
crossed product with a Galois group of simple structure, for instance abelian).
\end{remark}

\begin{remark}	
Over an arbitrary field $F$ an arbitrary
central simple algebra $\cA/F$ does not necessarily belong to 
one of the classes (i)-(iv). Hamilton in 1843
proved that over $\mathbb{R}$  not every central simple algebra is a matrix algebra. Albert
in 1932 (\cite{Al32}) gave an example of a 
non-cyclic division algebra, and later Amitsur in
1972 (\cite{Am72}) proved that there do exist noncrossed product algebras.
It is easily derived from the latter result that Amitsur's algebras
are not symbol algebras.
\end{remark}

Before we formulate our main results let us list some known facts with different properties $\mathcal{P}$.

(i) If $\mathcal{P}$ is a property to be a matrix algebra, then 
for a single finite dimensional central simple algebra $\cA$,
problem \ref{prob1} always has a solution, given by a so-called splitting field
for $\cA$. There are a lot of results about splitting fields,
which are regular extensions of the ground field, obtained by Witt,
Brauer, Roquette, Chatelet, Kovach, Heuser and others. 
In this case also problem \ref{prob2} has a solution: The
Severi-Brauer variety associated to $\cA$ has the required properties. 


(ii) If $\mathcal{P}$ is a property to be a crossed product algebra
with a given finite group $G$ and with injective restriction
homomorphism $\Br(F) \longrightarrow \Br(E)$, then the answer is also
positive. In the appendix to \cite[Th. 4]{BeFr01} 
a proof of an unpublished result of D. Saltman is given:

\begin{theorem}	 \label{t:Saltman}
Let $F$ be a field, $\cA$ a central simple algebra of degree $n$
over $F$ and $G$ a finite group of order $n$.  Then there exists a
finitely generated field extension $E/F$ such that:

$\cA_E$ is isomorphic to a $G$-crossed product;

the restriction homomorphism $\res: \Br(F)\longrightarrow 
\Br(E)$ is an injection.
\end{theorem}

\begin{remark} For $G$  cyclic and $\rho_n\in F$,
a simple proof of this fact is given in \cite[Th. 5.5.1]{GiSz}.
\end{remark}

%\begin{remark} For the case where $G$ is cyclic and $\rho_n\in F$,
%a simple proof  of this fact is given in \cite[Th. 5.5.1]{GiSz}.
%\end{remark}

%In the case where $\mathcal{P}$ is a property to be a symbol algebra of
%$\deg(A)$ if $F$ contains a primitive root of unity of degree
%$\deg(A)$ and $Br(F) \longrightarrow Br(E)$ is an injection the
%solution always exists and this is also due to Saltman \cite[Th. 5.5.1]{GiSz}.

\begin{remark} The first mentioning about existence of positive solutions in
the latter two cases is due to M. van den Bergh and A. Schofield.
They also showed how one could be able to prove corresponding
statements (\cite{BS92}).   In \cite{BS92} it was noted that given a central division algebra
$\cA$ over a field $F$, there exists a regular field extension
$E/F$ such that $\cA_E=\cA\otimes_F E$ is a cyclic division
algebra (see the discussion after Theorem 2.6 in  \cite{BS92}).
\end{remark}



%The first known central division algebras were cyclic. It is known
%that division algebras of degree 2, 3 and 6 are cyclic. But in
%case of degree 4 there are examples of noncyclic division
%algebras. Besides, $F$ is a local or global field, then any
%central division algebra is cyclic.

%Cyclic algebras are the most simple objects among central simple
%algebras over fields. Unfortunately, it is well known (see
%\cite{Am}) that not every such an algebra is cyclic. In view of this
%it seems that the following problem is of special interest.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In our paper we extend this result and prove that for a field $F$
there exists a regular field extension $E/F$ preserving indices of
all central simple $F$-algebras such that all $E$-algebras are cyclic.

As for Problem \ref{prob2}, we construct a quasi-affine variety such that the
existence of $L$-rational point of that variety is responsible
roughly speaking for $\cA \otimes_F L$ to be an algebra with a maximal
cyclic subfield coming from a given cyclic extension $Z/F$ of
degree $\deg(\cA)$. (The fact that for a given cyclic extension
$Z/F$ of degree $\deg(\cA)$ there exists at least one regular
extension $L/F$ such that $[LZ:L]=\deg(\cA)$ and $LZ$ is a maximal
subfield of $\cA\otimes_F L$ follows from \cite[Th. 3.7]{BeSc}.)

More precisely, in Section \ref{sec:cycl ext} we prove
the following 

\begin{theorem}	 \label{t:main new}
Let $F$ be a field.	 Then there exists a regular field extension
$E/F$ with the following properties:

%(i) for any central simple $k$-algebra $\cC$,
%$\cC\otimes_k E=\cC_E$ is cyclic,

(i) any central simple $E$-algebra is cyclic,

(ii) for any central simple $F$-algebra $\cC$,
$\ind(\cC_E)=\ind(\cC)$,

(iii) for any central simple $F$-algebra $\cC$,
$\exp(\cC_E)=\exp(\cC)$,

(iv) the restriction homomorphism $\res: \Br(F)\longrightarrow
\Br(E)$ is an injection.

\end{theorem}

%In the proof of this result we use arguments related to properties of
%Severi-Brauer varieties and one result of H. Miki on the embedding problem for cyclic extensions over fields
%with discrete valuations.

In Sections \ref{sec:symbol var} and \ref{sec:cycl var} we prove
the following 

\begin{theorem}	 \label{t:main symbol var}
Let $\rho_n\in F$ and $\cA$ be a central simple $F$-algebra of
degree $n$. Then there exists a quasi-affine $F$-variety
$\Symb(\cA)$ such that for a field extension $L/F$ $\Symb(\cA)$ has
an $L$-rational point iff $\cA_L$ is a symbol algebra.
\end{theorem}

\begin{theorem}	 \label{t:main cycle var}
Let $\cA$ be a central simple algebra over a field $F$ of degree
$n$ and $K/F$ a cyclic field extension of degree $n$. Then there exists a 
quasi-affine $F$-variety $C(\cA, K)$ such that for a
field extension $L/F$ with the property $[K L:L]=[K:F]$ $C(\cA,
K)$ has an $L$-rational point iff $K L$ is a maximal subfield of
$\cA_L$.
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To illustrate the usefulness of our considerations we will show how
one can reduce a well known conjecture of Suslin related to generic
elements of reduced Whitehead groups to the case of central division
algebras of a special type. 

First recall a few definitions.
For any central simple algebra $\cA/F$ a reduced norm mapping $Nrd:
\cA\longrightarrow F$ is defined. Its restriction to the
multiplicative group $\cA^*$ of $\cA$ gives a homomorphism $Nrd: \cA^*
\longrightarrow F^*$. Then the factor group $SK_1(\cA)$ of the kernel
of this homomorphism by the commutator subgroup of $\cA^*$ is usually
called the reduced Whitehead group of $\cA$. At least two observations
make this group of a special interest.  The first is its well known
relation to Kneser-Tits conjecture for algebraic groups (\cite{Gi07})
and second its connection
with a problem of rationality for such groups via group of
$R$-equivalences classes for a group algebraic varieties (\cite{Gi07}).

In this context A.A. Suslin assumed that

\begin{conjecture} (Suslin, 1991, \cite{Su91}, \cite{Su06}). 
Let $\cA/F$ be a central simple algebra of $\ind(\cA)$ prime to
$char(F)$, $\mathbb{G}$ the algebraic group defined by $SL(1,\cA)$,
$F(\mathbb{G})$ its function field, $\mathbb{G}(F(\mathbb{G}))$ the
group of $F(\mathbb{G})$-rational points of $\mathbb{G}$. If
$\ind(\cA)$ is not square-free then a generic point $\xi \in
\mathbb{G}(F(\mathbb{G}))$	leads to a nontrivial generic element of
$SK_1(\cA\otimes_F F(\mathbb{G}))$.
\end{conjecture}

Nowadays this conjecture is proved only in case where $\ind(\cA)$ is
divisible by 4 (\cite{Me93}).

\begin{remark} It is easy to see that to prove Suslin's conjecture
it is sufficient to prove the following statement: for any central
simple algebra $\cA/F$ with $\ind(\cA)$ which is not square-free there
exists a field extension $E/F$ such that $SK_1(\cA\otimes_F E)\ne \{0\}$. 
\end{remark}

Using
above results we prove immediately the following.


\begin{theorem} Suslin's conjecture is true iff it is true for all cyclic
division algebras.
\end{theorem}

Combining the above theorems with the main result of \cite{PrTiYa08}
one can prove the stronger 

\begin{theorem} Suslin's conjecture is true iff it is true for all cyclic
division algebras of the form $(a,c)_p\otimes (b,d)_p$.
\end{theorem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cyclicity after a scalar extension} \label{sec:cycl ext}

In order to prove the Theorem \ref{t:main new} we need a few
preliminary statements.


\begin{prop}{\textrm{(\cite[Th. 1.3]{BS92}, \cite[Th. 13.10]{Sa99})}}
 Let $\cD$, $\cE$ be central
division algebras over $F$ of indices $m$ and $n$ respectively. Let
$BS(\cE)$ be the Severi-Brauer variety of $\cE$ and let $K$ be its
function field. Then
$$
\ind(\cD\otimes_F K)= \gcd\{ \ind(\cD\otimes_F \cE^i)\}
$$
where $i$ ranges from 1 to $n$.
\end{prop}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{remark} In the literature the latter formula is called the
index reduction formula.
\end{remark}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{corol}  \label{c:coprime}
 Let $\cD$, $\cE$ be central
division algebras over $F$. Let $K$ be the function field of the
Severi-Brauer variety $BS(\cE)$. Assume that $\ind (\cD)$ is
coprime to $\ind (\cE)$. Then $ \ind(\cD\otimes_F K)= \ind(\cD)$.
\end{corol}

\noindent {\it Proof}. Use the index reduction formula. \qed

\bigskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{lemma} {\textrm{(\cite[p. 176, ex. 7]{Bou})}} \label{l:Bourbaki}
Let $E/F$ be a cyclic field extension of degree $q^{l-1}$ ($q$ is
prime). Let $\Gal(E/F)=<\sigma>$. Let also $(\char(F),q)=1$. Assume
that there exists an element $\beta\in E$ such that
$N_{E/F}(\beta)=\rho_q$ ($\rho_q$= primitive $q$-th root of unity).
Let $a\in E$ be such that $a^{\sigma}/a=\beta^q$. Then

(i) for any $\lambda\in F^*$ the polynomial
$$
x^q-\lambda a
$$
is irreducible over $E$,

(ii) if $\theta$ is a root of that polynomial, then $E(\theta)$ is
a cyclic extension of $F$ of degree $q^l$.
\end{lemma}


\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The following result we will use only in the case $p=2$,
but for the sake of generality, we will prove it for any $p$.


\begin{lemma} \label{l:extension prime index}
%Let $\cA/k$ be a central simple algebra, $\ind(\cA)=p^n$.
Assume $\rho_p\in F$. Then for any $m\in \mathbb{N}$ there exists
a tower of field extensions
$$
F\subset K \subset E
$$
such that

\begin{enumerate}
\item[\text{\rm{(i)}}] $E/F$ is a regular extension;
\item[\text{\rm{(ii)}}] $E/K$ is a cyclic extension of degree
$p^m$;
%\item[\text{\rm{(iii)}}] $\ind(\cA_F)=\ind(\cA)$;
\item[\text{\rm{(iii)}}] for any central simple $F$-algebra $\cC$,
%with $(\ind(\cC),p)=1$
 $\ind(\cC_E)=\ind(\cC)$.
\end{enumerate}
\end{lemma}

\noindent {\it Proof}. %Since $|k(\rho_p):k|$ is coprime to $p$,
%without loss of generality we can assume that $\rho_p\in k$.

We will prove the statement by induction on $m$. Let $\cC$ be a
central simple $F$-algebra. % $\cC$ with $(\ind(\cC),p)=1$.
If $m=1$, set $K=F(x)$ and $E=K(\sqrt[p]{x})$ where $x$ is a
transcendental variable.

Suppose that the statement of lemma is true for $m=m_0$, i.e. there
exists a tower of field extensions $F\subset K_{m_0}\subset E_{m_0}$
such that $E_{m_0}/K_{m_0}$ is cyclic with
$|E_{m_0}:K_{m_0}|=p^{m_0}$, $E_{m_0}/F$ is regular and
$$
\ind(\cC_{E_{m_0}})=\ind(\cC).
$$

Consider the case $m=m_0+1$. Set
$$
\cB=(E_{m_0}/K_{m_0},\sigma, \rho_p)
$$
be a cyclic algebra over $K_{m_0}$. Let $M$ be the function field
of the corresponding Severi-Brauer variety $\SB(\cB)$. %Using
%corollary \ref{c:coprime} we conclude that
%$\ind(\cC_E)=\ind(\cC)$.

Note that the compositum $M E_{m_0}/M$ is a cyclic extension of
degree $p^{m_0}$. Since $E_{m_0}$ is a splitting field of $\cB$,
then $M E_{m_0}$ is a purely transcendental extension of
$E_{m_0}$. Hence $M E_{m_0}/F$ is a regular extension. Moreover,
$$
\ind(\cC_{M E_{m_0}})=\ind(\cC_{ E_{m_0}})=\ind(\cC).
$$

Besides, $\cB_M\sim 1$. Then there exists $\beta\in M E_{m_0}$
such that
$$
N_{M E_{m_0}/M}(\beta)=\rho_p.
$$
Let $a\in M E_{m_0}$ be such that $a^{\sigma}/a=\beta^p$. Let
also $y$ be a new transcendental variable. Then $M
E_{m_0}(y)/M(y)$ is cyclic of degree $p^{m_0}$. Moreover, using
Lemma \ref{l:Bourbaki} we conclude that
$$
M E_{m_0}(\sqrt[p]{ay})/M(y)
$$
is cyclic of degree $p^{m_0+1}$. It is clear that $M
E_{m_0}(\sqrt[p]{ay})/F$ is regular. Finally,
$$
\ind(\cC_{M E_{m_0}(\sqrt[p]{ay})})= \ind(\cC_{M
E_{m_0}})=\ind(\cC)
$$
 \qed

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{lemma} \label{l:extension any index}
Let $F$ be a field. Then for any $n\in \mathbb{N}$ there exists a
tower of field extensions $F\subset K\subset E$ such that $E/F$ is
regular, $E/K$ is cyclic of degree $n$ and
$$
\ind(\cC_E)=\ind(\cC)
$$
for any central simple $F$-algebra $\cC$.
\end{lemma}

\bigskip

\noindent {\it Proof}. At first, consider the case where
$\char(F)=2$ or $4 \nmid n$. Let $x$ be a transcendental variable
and $v_x$ the valuation of $F(x)$ corresponding to the polynomial
$x$. Then by \cite[Th. 5]{Mi78}, there exists a cyclic field extension
$E/F(x)$ of degree $n$ such that the completion $E_{v_x}$ coincides
with $F(x)_{v_x}$. Note that $F(x)_{v_x}=F((x))$. Hence for any central simple
$F$-algebra $\cC$, $\ind (\cC)=\ind
(\cC_{F((x))})=\ind(\cC_{E_{v_x}})$. Then $\ind (\cC)=\ind (\cC_E)$.

Moreover, since $F$ is algebraically closed in $E_{v_x}$, then
$E/F$ is a regular extension.

Now consider the case $\char(F)\ne 2$ and $4 | n$. Let $n=2^s m$
where $2\nmid m$. In view of the case considered above, there
exists a tower of field extensions $F\subset K_1\subset E_1$ such
that $E_1/F$ is regular, $E/K$ is cyclic of degree $m$ and $E_1$
preserves indices of $F$-algebras. Besides, by Lemma
\ref{l:extension prime index}, there exists a tower of field
extensions $F\subset K_2\subset E_2$ such that $E_2/F$ is a regular
extension, $E_2/K_2$ is cyclic of degree $2^s$ and
$\ind(\cC_{E_2})=\ind(\cC)$ for any central simple $F$-algebra
$\cC$. Set
$$
K=K_1  K_2, \;\;\; E=E_1 E_2
$$
be respectively the free composita over $F$ of $K_1, K_2$ and
$E_1, E_2$. Then  $E/K$ is a cyclic extension of degree $n$.
Furthermore, since $E_i$ does not change the index of $\cC$, then
$$
\ind(\cC)=\ind(\cC_{E_1})=\ind(\cC_{E_1 E_2})=\ind(\cC_E).
$$
\qed

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%{\bf Remark.} If not all $\rho_{p_i}\in k$, the latter lemma is
%also true provided that $(|k(\rho_{p_i}):k|,p_j)=1$ for all
%$\rho_{p_i}\not \in k$ and $j=1,s,s$.
%
%\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{lemma} \label{l:indices_tensor_product}
Let	 $\cA$ and $\cB$ be central simple $F$-algebras. Assume $\ind
(\cA)=p^m$, $\ind (\cB)=p^n$ and $m\ge n$. Then $\ind
(\cA\otimes_F \cB)\ge p^{m-n}$.
\end{lemma}

\noindent {\it Proof}. Let $E/F$ be a field extension of degree
$p^n$ which splits $\cB$. Let also $\ind (\cA\otimes_F \cB)=p^s$.
Assume $p^s< p^{m-n}$. Then there exists a field extension $L/F$
of degree $p^s$ splitting $\cA\otimes_F \cB$. Hence
$$
1\sim (\cA\otimes_F \cB)_{E L}\sim \cA_{E L}\otimes_{E L} \cB_{E
L}\sim \cA_{E L}.
$$
Thus $E L$ is a splitting field of $\cA$. Since $|E L:F|<p^m$,
then $\ind (\cA)<p^m$. Contradiction. \qed

\bigskip


\begin{lemma} \label{l:index p-th}
Let	 $\cA$ be a central simple $F$-algebra with $\ind (\cA)=p^m$.
Then $\ind (\cA^{\otimes p})<\ind (\cA)$.
\end{lemma}

\noindent {\it Proof}. Without loss of generality we can assume
that there exists a splitting field $L$ of $\cA$ such that
$|L:F|=\ind(\cA)$ and $L$ contains a subfield $K$ with $|L:K|=p$.
%(\cite[???]{Pi}.
Then $\ind (\cA_K)=p$. Hence $1=\ind (\cA^{\otimes p}_K)$. Thus
$\ind (\cA^{\otimes p})\le |K:F| < |L:F|= \ind (\cA)$. \qed

\bigskip

\begin{lemma} \label{l:dv_division_algebra}
Let $K/F$ be a cyclic field extension,
$\<\sigma_i\>=\Gal(K(z)/F(z))$ and $z$ is a transcendental variable.
Let also  $\cC$ be a central division $F$-algebra such that
$\cC_K$ is a division algebra. Then
$$
(K(z)/F(z),\sigma,z)\otimes \cC_{F(z)}
$$
is a division $F(z)$-algebra.
\end{lemma}

\noindent {\it Proof:}  Analogous to that of Proposition 1.3
from \cite{KuRoTiYa}. \qed

\bigskip

In the notations of the previous lemma we have immediately the
following

\begin{corol} \label{c:l:dv_division_algebra}
Let	 $\cA$ be a central simple $F$-algebra such that
$\ind(\cA_K)=\ind(\cA)$. Then
$$
\ind\left((K(z)/F(z),\sigma,z)\otimes \cA_{F(z)}\right)= \ind
((K(z)/F(z),\sigma,z)) \ind (\cA).
$$
\end{corol}

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{theorem}	 \label{t:main}
Let $\cA$ be a central simple algebra over a field $F$. Then there
exists a regular field extension $M/F$ such that 
%$\cA_M$ is cyclic
%and $\ind(\cC_M)=\ind(\cC)$ for any central simple $F$-algebra
%$\cC$. 

(i) $\cA_M$ is cyclic,

(ii) for any central simple $F$-algebra $\cC$,
$\ind(\cC_M)=\ind(\cC)$,

(iii) for any central simple $F$-algebra $\cC$,
$\exp(\cC_M)=\exp(\cC)$,

(iv) the restriction homomorphism $\res: \Br(F)\longrightarrow
\Br(M)$ is an injection.

Moreover, for any field extension $L/F$, the free compositum
$M L$ over $F$ preserves indices of $L$-algebras.
\end{theorem}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent {\it Proof}. Let $\deg(\cA)=n=p_1^{n_1}\dots p_s^{n_s}$
($p_i$-s are distinct primes) and $\cA=\otimes_{i=1}^s \cA_i$, where
$\ind(\cA_i)=p_i^{l_i}$, $l_i\le n_i$. By Lemma \ref{l:extension any
index}, there exists a tower of field extensions $F\subset K \subset
E$ such that $E/F$ is regular, $E/K$ is a cyclic extension of degree
$n$ and $E$ preserves indices of $F$-algebras. Let $E_i/K$ be a
cyclic subextension of degree $p_i^{n_i}$.

Consider cyclic algebras
$$
\cD_i=(E_i(z)/K(z), \sigma_i, z),\; i=1,\dots,s,
$$
where $<\sigma_i>=\Gal(E_i(z)/K(z))$ and $z$ is a transcendental
variable.

%Note that $\ind(\cD_i)=\exp(\cD_i)=p_i^{n_i}$. Moreover,
%$p_i^{n_i}|\exp(\cD_i^j\otimes_{K(z)} \cC_{K(z)})$ for any central
%simple $K$-algebra $\cC$ and $j$ not divisible by $p$. Indeed,
%consider the ramification map
%$$
%\ram: \Br\;K(z) \longrightarrow H^1(\Gal(\overline K/K,\Q/\Z)
%$$
%corresponding to the discrete valuation of $K(z)$ defined by $z$
%(see e.g. \cite[Ch. 10]{Sal}). Then the order of
%$\ram([(\cD_i^j\otimes_{K(z)} \cC_{K(z)})])$ is $p_i^{n_i}$. Hence
%$p_i^{n_i}|\exp(\cD_i^j\otimes_{K(z)} \cC_{K(z)})$.

Set
$$
\cD=\otimes_{i=1}^s \cD_i.
$$
Since
$$
\cD_i\sim (E(z)/K(z), \sigma, z^{n/p^{n_i}}),
$$
where $<\sigma>=\Gal(E(z)/K(z))$, then
$$
\cD\cong (E(z)/K(z), \sigma, z^{\Sigma_{i=1}^s n/p^{n_i}}).
$$
Then $\cD$ is cyclic of index $n$ with a maximal subfield $E(z)$.

One has
$$
\cD\sim \cD\otimes_{K(z)} \cA_{K(z)}^{op}\otimes_{K(z)}
\cA_{K(z)}.
$$
Let $M$ be the function field of the Severi-Brauer variety
$\SB(\cD\otimes_{K(z)} \cA_{K(z)}^{op})$. Then $\cA_M\sim \cD_M$.
Since $\deg(\cA_M)=\deg(\cD_M)$, then $\cA_M\cong \cD_M$

Let $\cC$ be a central simple  $F$-algebra and $\cC=\otimes_{i=1}^m
\cC_i $ the decomposition of $\cC$ as a tensor product of algebras
of primary indices. Since $\ind(\cC_M)=\prod_{i=1}^m
\ind({\cC_i}_M)$, then it is enough to consider the case where $\cC$
has a primary index. Moreover, if $p_i \nmid \ind(\cC)$, $1\le i\le
s$, then $\ind(\cC_M)=\ind(\cC)$ by Corollary \ref{c:coprime}. Thus
we will assume that $\ind(\cC)=p_i^{m_i}$ is a power of $p_i$ for
some $1\le i\le s$.

Using the index reduction formula we obtain
$$
\ind(\cC_M) = \gcd \{\ind(\cD^{\otimes j}\otimes_{K(z)}
{\cA_{K(z)}^{op}}^{\otimes j}\otimes_{K(z)} {\cC_{K(z)}})\}
$$
where $j$ ranges from 1 to $n$.

Thus since $\ind(\cC)$ is a power of some $p_i$, then
$$
\ind(\cC_M)=\min_{j=1}^{n_i} \{\ind(\cD_i^{\otimes
j}\otimes_{K(z)} {{\cA_i}_{K(z)}^{op}}^{\otimes j}\otimes_{K(z)}
{\cC_{K(z)}})\}.
$$

Consider the algebra $\cB_{i,j}=\cD_i^{\otimes j}\otimes_{K(z)}
{{\cA_i}_{K(z)}^{op}}^{\otimes j}\otimes_{K(z)} {\cC_{K(z)}}$. Note
that by Corollary \ref{c:l:dv_division_algebra},
$$
\ind(\cB_{i,j})=\ind(\cD_i^{\otimes j})
\ind({{\cA_i}_K^{op}}^{\otimes j}\otimes_K {\cC_K}).
$$
Fix some $j$. Let $j={p_i}^t j_1$, where $p_i\nmid j_1.$ Then
$\ind(\cD_i^{\otimes j})=p_i^{n_i-t}$. Let $\ind({\cA^{op}}^{\otimes
j})=p_i^{s_i}$. Then $s_i\le l_i-t$ by Lemma \ref{l:index p-th}.
Hence
$$
\ind(\cB_{i,j})=p_i^{n_i-t} p_i^{|s_i-m_i|}=p_i^{n_i-t+|s_i-m_i|}
$$
in view of Lemma \ref{l:indices_tensor_product}.


 Finally consider two cases. If $s_i\ge m_i$,
then $n_i-t \ge m_i$ and $n_i-t+|s_i-m_i|\ge m_i$. If $s_i< m_i$,
then $n_i-t+|s_i-m_i|= n_i-t-s_i+m_i\ge m_i$. Therefore, $
\ind(\cB_{i,j})\ge p^{m_i}=\ind(\cC)$ for any $j$. Thus
$\ind(\cC_M)=\ind(\cC)$.

%%%%%%%%%%%%%%%%%%%%%%%%

Note that, for a field extension $M/F$,  preserving 
indices for all $F$-algebras implies also preserving
exponents of $F$-algebras. Indeed, assume $\cC_M^{\otimes m}\sim
1$ for some central simple $F$-algebra $\cC$. Since
$$
1=\ind(\cC_M^{\otimes m})=\ind(\cC^{\otimes m}),
$$ 
then
$\cC^{\otimes m}\sim 1$. Thus $\exp(\cC_M)=\exp(\cC)$. Moreover,
preserving exponents implies, in turn, that the restriction
homomorphism
$$
\res:\Br(F)\longrightarrow \Br(M)
$$
is an embedding. 


%%%%%%%%%%%%%%%%%%%%%%%%

Now consider the last statement about free composita. The free
compositum $L M$ can be constructed using the same procedure as
the field $M$. We just replace the constant field $F$ by $L$. Now
the statement about preserving indices of $L$-algebras is
obtained automatically.
 \qed
\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{remark} 
Note that preserving exponents for all $F$-algebras does not
imply preserving indices as is easily seen. 
\end{remark}

%% \bigskip


%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%% \begin{example} In this example we will construct a field
%% extension preserving exponents, but not preserving indices of
%% central simple algebras.

%% Let $F$ be a field such that there exists a central simple algebra
%% $\cA$ with $\ind(\cA)=4$ and $\exp(\cA)=2$. Let also $x$ and $y$
%% be transcendental variables. Set
%% $$
%% \cD=(x,y)\otimes_{F(x,y)} \cA^{op}_{F(x,y)}.
%% $$
%% Let $M$ be the function field of the Severi-Brauer variety
%% corresponding to $F(x,y)$-algebra $\cD$. Using the index
%% reduction formula it is easy to prove that $E$ does not reduce
%% exponents of $F$-algebras. On the other hand, $\cA_M\sim
%% (x,y)_M$. Hence, $2=\ind(\cA_M)<\ind(\cA)$.

%% \end{example}

\bigskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%{\bf Remark.} If not all $\rho_{p_i}\in k$, theorem is also true
%provided the same conditions as in the remark after Lemma
%\ref{l:extension any index} are satisfied.
%
%\bigskip
%
%\begin{corol}
%Let $\cA$ be a central simple algebra over a field $k$,
%$\ind(\cA)=p^n$ ($p$ is prime). Then there exists a field
%extension $E/k$ such that $\cA_E$ is cyclic and
%$\ind(\cA_E)=\ind(\cA)$.
%\end{corol}
%
%\noindent {\it Proof}. Since $|k(\rho_p):k|$ is coprime to $p$,
%without loss of generality we can assume that $\rho_p\in k$. Now
%the statement follows from the theorem. \qed
%
%\bigskip



%\begin{theorem} \label{t:multiple}
%Let $\{\cA_{\alpha}\}_{\alpha\in I}$ be a family of central simple algebras over a field
%$k$. Then there exists a regular field extension $E/k$
%such that all the algebras ${\cA_{\alpha}}_E$ are cyclic, and for any
%central simple $k$-algebra $\cC$,
%$\ind(\cC_E)=\ind(\cC)$ .
%\end{theorem}

Now we are in a position to prove the main result.

\noindent {\it Proof of Theorem \ref{t:main new}}.

Note that in view of Remark after Theorem \ref{t:main},
preserving indices implies preserving exponents and injectivity of
the restriction map.

First, we will prove that for any field $K$ there exists a regular
field extension making all $K$-algebras cyclic and preserving their
indices. If the set $I$ of central simple $K$-algebras is finite,
use Theorem \ref{t:main} and induction.

If $I$  is countable, 
we can construct a sequence of field extensions
$$
E_1\subset E_2 \subset \dots E_{i} \subset E_{i+1} \subset \dots
$$
such that $E_{i+1}$ is a regular extension of $E_i$ making ${\cA_{i+1}}_{E_i}$ cyclic and
preserving indices of $E_i$-algebras. Then the field $E=\cup_i E_i$ has the required properties.

Finally, if $I$ is not countable, then the statement can be proved using Zorn's lemma.
Indeed, consider the set

$$
\cM=
\left\{
\begin{array}{l|l}
&  \mbox{for any field extension $L/F$, the free}\\
\mbox{regular field} & \mbox{compositum $L K$ over $F$ preserves} \\
\mbox{extensions }K/F & \mbox{indices of $L$ -algebras and there exists}\\
& \mbox{an $F$-algebra } \cA \mbox{ s.t. $\cA_K$ is
cyclic}
\end{array}
\right\}
$$
All the fields in $\cM$ are assumed to be in some universal
domain. Inclusions of fields define partial order on this set. If
we have a totally ordered subset $\cS=\{K_{\alpha}\}\subset \cM$,
then $\cup_{\alpha}K_{\alpha}$ is an upper bound for this subset.
Hence, by Zorn's lemma, there exists a maximal element $E$ in
$\cM$.

We will show that $E$ has the required properties by the rule of
contraries. Assume that there exists an $F$-algebra $\cA$ such that
$\cA_E$ is not cyclic. By Theorem \ref{t:main}, there exists a field
extension $E_1\in \cM$ such that $\cA_{E_1}$ is cyclic. Then $E_1 E
\in \cM$ and contains $E$. Contradiction. Thus, the field $E$ has the
required properties.

Now we will finish the proof of the theorem. Let $K_0$ be the
field making all $F$-algebras cyclic and preserving their
indices. Let also $K_1$ be the field making all $K_0$-algebras
cyclic and preserving their indices, and so on. Then the field
$\cup_i K_i$ has the required properties.

\qed

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symbol algebra variety $\Symb(\cA)$} \label{sec:symbol var}

Recall firstly how the Severi-Brauer variety corresponding to a
central simple algebra $\cA$ can be defined by polynomial
equations. For a given $n$-dimensional $F$-vector space $V$, let
$\Grass_F(m,V)$ be the set of its $m$-dimensional subspaces.
$\Grass_F(m,V)$ has the structure of a projective variety via the
Pl\"{u}cker embedding
$$
\Grass_F(m,V) \longrightarrow {\mathbb{P}}(\wedge^m V),
$$
$$
Fw_1\oplus Fw_2\oplus \dots \oplus Fw_m \mapsto F(w_1\wedge w_2
\wedge\dots \wedge w_m),
$$
where $\wedge^m V$ is the $m$-th wedge power of $V$. Fixing a
base $e_1,e_2,\dots,e_n$ for $V$ we obtain a base $e_{i_1}\wedge
e_{i_2} \wedge\dots \wedge e_{i_m}$, $1\le i_1 <i_2<\dots < i_m\le n$ for
$\wedge^m V$. This gives in turn homogeneous coordinates for
$F(w_1\wedge w_2 \wedge\dots \wedge w_m)$.

Now let $\cA$ be a central simple $F$-algebra with $F$-base
$e_1,e_2,\dots,e_{n^2}$.
%Then $\dim_F \cA=n^2$ for some integer $n$.
Then the Severi-Brauer variety $\SB(\cA)$ is a subvariety of the
Grassmanian $\Grass_F(n,\cA)$ consisting of those points which
correspond to right ideals of $\cA$. This condition can be expressed
by polynomial relations as follows (see \cite[p.112]{Ja96}).)

Let $T\subset \cA^*$ be a subset such that the multiplicative
group generated by $T$ contains a base of $\cA$. 
%Let $I\in
%\Grass_F(n,\cA)$ corresponds to a vector subspace $Fw_1\oplus
%Fw_2\oplus \dots \oplus Fw_m\subset \cA$, i.e. $I=F(w_1\wedge w_2
%\wedge\dots \wedge w_n)$. Then $I$ is a right ideal of $\cA$ iff
%$I=F(w_1 t\wedge w_2t \wedge\dots \wedge w_n t)$ for all $t\in T$.
Let $I= F(w_1\wedge w_2
\wedge\dots \wedge w_n) \in  \Grass_F(n,\cA)$, i.e., $I$
corresponds to a vector space $W=Fw_1\oplus
Fw_2\oplus \dots \oplus Fw_n\subset \cA$. 
Then $W$ is a right ideal of $\cA$ iff
$I=F(w_1 t\wedge w_2t \wedge\dots \wedge w_n t)$ for all $t\in T$.







For $t\in T$, write
\begin{equation} \label{eq:basis}
e_i t=\sum a_{ij}e_j.
\end{equation}
Then
$$
e_{i_1}t\wedge e_{i_2}t \wedge\dots \wedge e_{i_n}t=\sum_{1\le j_1
<\dots<j_n\le n^2} t_{j_1,\dots,j_n,i_1,\dots,i_n} e_{j_1}\wedge
e_{j_2} \wedge\dots \wedge e_{j_n},
$$
where
$$
t_{j_1,\dots,j_n,i_1,\dots,i_n}=
 \left|
\begin{array}{ccc}
a_{i_1j_1}& \dots & a_{i_1j_n}\\
\vdots & & \vdots \\
a_{i_nj_1}& \dots & a_{i_nj_n}\\
\end{array}
\right|.
$$
Then for
$$
w_1\wedge w_2 \wedge\dots \wedge w_n=\sum_{1\le i_1 <\dots<i_n\le
n^2} p_{i_1,\dots,i_n} e_{i_1}\wedge e_{i_2} \wedge\dots \wedge
e_{i_n}
$$
we have
$$
w_1 t\wedge w_2t \wedge\dots \wedge w_nt=\sum_{1\le i_1
<\dots<i_n\le n^2} q_{i_1,\dots,i_n} e_{i_1}\wedge e_{i_2}
\wedge\dots \wedge e_{i_n},
$$
where
$$
q_{i_1,\dots,i_n}=\sum_{1\le j_1 <\dots<j_n\le n^2}
t_{i_1,\dots,i_n,j_1,\dots,j_n}p_{j_1,\dots,j_n}.
$$

Thus $F(w_1 \wedge w_2 \wedge\dots \wedge w_n)=F(w_1 t\wedge w_2t
\wedge\dots \wedge w_n t)$ iff their Pl\"{u}cker coordinates are
proportional, that is
$$
q_{i_1,\dots,i_n}p_{j_1,\dots,j_n}
=q_{j_1,\dots,j_n}p_{i_1,\dots,i_n}.
$$

Hence each $t\in T$ defines the following set of polynomial
equations for $\SB(\cA)$
$$
\left(\sum_{1\le k_1 <\dots<k_n\le
n^2}t_{i_1,\dots,i_n,k_1,\dots,k_n}\xi_{k_1,\dots,k_n}\right)
\xi_{j_1,\dots,j_n}
$$
$$-\left(\sum_{1\le k_1 <\dots<k_n\le
n^2}t_{j_1,\dots,j_n,k_1,\dots,k_n}\xi_{k_1,\dots,k_n}\right)
\xi_{i_1,\dots,i_n}=0
$$
where $1\le i_1 <\dots<i_n\le n^2$,	 $1\le j_1 <\dots<j_n\le n^2$.

Thus the Severi-Brauer variety of $\cA$ is defined in
$\Grass_F(n,\cA)$ by this system of equations for all $t\in T$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let $\rho_n\in F$ and $\cB$ be a central simple $F$-algebra of
degree $m$. Set
$$
\cD=(x,y)_n\otimes\cB_{F(x,y)}
$$
where $F(x,y)$ is a purely transcendental extension of $F$. Let
$$
\sqrt[n]{x}^i \sqrt[n]{y}^j, \;0\le i,j\le n-1,
$$
be a standard base for the algebra $(x,y)_n$ and
$v_1,\dots,v_{m^2}$ a base for $\cB$ over $F$ consisting from
invertible elements. Then
$$
 \sqrt[n]{x}^i \sqrt[n]{y}^j v_l, \;0\le i,j\le n-1, \;1\le
l\le m^2,
$$
is a base for $\cD$ over $F(x,y)$.

 Now using the procedure above we
will construct equations for the Severi-Brauer variety $\SB(\cD)$.
First of all take standard polynomials $G_j\in
F(x,y)[\xi_0,\dots,\xi_N]$, $j\in J$, defining the Grassmanian
$$
\Grass(nm, \cD)\subset \PP^N,
$$
where $N=C_{n^2m^2}^{nm}-1$. Note that
coefficients of $G_j$-s, $j\in J$, belong to the set $\{\pm 1, 0\}$.

Further, set
$$
T_{\cD}=\{ \sqrt[n]{x}, \sqrt[n]{y}, v_1,\dots, v_{m^2}\}.
$$
Then the multiplicative group generated by $T_{\cD}$ contains a
base of $\cD$ over $F(x,y)$. Then each $t\in T_{\cD}$ defines a
family of polynomial equations $\{F^{(t)}_i\}$, $i\in I_t$, of
$\SB(\cD)$. Note that even $\{F^{(t)}_i\}\in
F[x,y][\xi_0,\dots,\xi_N]$, $i\in I_t$. To see this it is enough
to show that, for each $t\in T_{\cD}$, the coefficients $a_{ij}$ from
(\ref{eq:basis}) belong to $F[x,y]$. Indeed, if $t=v_l$, then
$$
(\sqrt[n]{x}^i\sqrt[n]{y}^j v_k)v_l= \sqrt[n]{x}^i\sqrt[n]{y}^j
(\sum a_s v_s)
$$
for some $a_s\in F$. If $t=\sqrt[n]{x}$, then

\begin{equation} \label{eq:expansion}
(\sqrt[n]{x}^i\sqrt[n]{y}^j v_k)\sqrt[n]{x}= \left\{
\begin{array}{cc}
\rho_n^j\sqrt[n]{x}^{i+1}\sqrt[n]{y}^j v_k& \hbox{ if  } i<n-1;\\
x\rho_n^j\sqrt[n]{y}^j v_k& \hbox{ if	} i=n-1.
\end{array}
\right.
\end{equation}

Similar relations hold for $\sqrt[n]{y}$.

Now let $L/F$ be a field extension and $a,b\in L^*$. Consider the
algebra
$$
\overline{\cD}=(a,b)_n\otimes_L \cB_L.
$$
Then
$$
 \sqrt[n]{a}^i \sqrt[n]{b}^j v_l, \;0\le i,j\le n-1, \;1\le
l\le m^2,
$$
is a base for $\overline{\cD}$ over $L$. Moreover, the
multiplicative group generated by $\{ \sqrt[n]{a}, \sqrt[n]{b},
v_l| \;1\le l\le m^2\}$	 contains a base of $\overline{\cD}$ over
$L$. Hence this set can be used in order to construct equations of
the Severi-Brauer variety of $\SB(\overline{\cD})$.



Note that 
$$
(\sqrt[n]{a}^i\sqrt[n]{b}^j v_k)\sqrt[n]{a}= \left\{
\begin{array}{cc}
\rho_n^j\sqrt[n]{a}^{i+1}\sqrt[n]{b}^j v_k& \hbox{ if  } i<n-1;\\
a\rho_n^j\sqrt[n]{b}^j v_k& \hbox{ if	} i=n-1.
\end{array}
\right.
$$
Thus the coefficients  of this expansion are obtained as the specialization $x\mapsto a$, $y\mapsto b$ of
the coefficients of the expansion \ref{eq:expansion}.
%Note that the coefficients $\alpha_{lij}^{(s,q)}$ of the expansion
%$$
%(\sqrt[n]{a}^s\sqrt[n]{b}^q v_q)\sqrt[n]{a}=\sum
%\alpha_{lij}^{(s,q)}\sqrt[n]{a}^i\sqrt[n]{b}^jv_l
%$$
%are obtained as the specialization $x\mapsto a$, $y\mapsto b$ of
%coefficients $a_{lij}^{(s,q)}$ of the expansion
%$$
%(\sqrt[n]{x}^s\sqrt[n]{y}^q v_q)\sqrt[n]{x}=\sum
%a_{lij}^{(s,q)}\sqrt[n]{x}^i\sqrt[n]{y}^jv_l
%$$
Similar relations take place for $\sqrt[n]{b}$ and $v_l$, $1\le
l\le m^2$.

Hence polynomials from $L[\xi_1,\dots,\xi_N]$ defining
$\SB(\overline{\cD})$ are obtained as the specialization
$$
x\mapsto a,\; y\mapsto b
$$
of polynomials in $F[x,y][\xi_1,\dots,\xi_N]$ defining
$\SB({\cD})$.

Thus we have proved the following

\begin{lemma}  \label{l:spec1}
Let $\rho_n \in F$, $\cB$ a central simple
$F$-algebra of degree $m$ and $a,b\in L^*$. Let also
$\cD=(x,y)_n\otimes\cB_{F(x,y)}$.
Then the Severi-Brauer
variety $\SB(\cD)$ of $\cD$ can be defined by polynomials in
$F[x,y][\xi_0,\dots,\xi_N]$ ($N=C_{n^2m^2}^{nm}-1$) such that
for any field extension $L/F$ and $a,b\in L^*$, 
their specialization $x\mapsto a$, $y\mapsto b$ gives polynomials
in $L[\xi_0,\dots,\xi_N]$ defining the Severi-Brauer variety
$\SB(\overline{\cD})$ of 
$\overline{\cD}=(a,b)_n\otimes\cB_L$. 
\end{lemma}

Now we are in a position to prove the main result of this section.

%\begin{theorem}  \label{t:main} Let $\rho_n\in F$ and $\cA$ be a
%central simple $F$-algebra of degree $n$. Then there exists a
%quasi-affine $F$-variety $\Symb(\cA)$ such that for a field
%extension $L/F$ $\Symb(\cA)$ has an $L$-rational point iff $\cA_L$
%is a symbol algebra.
%\end{theorem}

\noindent {\it Proof of Theorem \ref{t:main symbol var}}. Set
$$
\cD=(x,y)_n\otimes\cA^{op}_{F(x,y)}
$$
where $F(x,y)$ is a purely transcendental extension of $F$.

The Severi-Brauer variety of $\cD$ is a subvariety of the
projective space $\PP_{F(x,y)}^N$, where $N=C_{n^2}^n-1$. Let
$F_j\in F[x,y][\xi_0,\dots,\xi_N]$, $j\in J$, be polynomials
defining $\SB(\cD)$ constructed in the proof of Lemma
\ref{l:spec1}. Now consider the $F_j$ as polynomials in
$F[x,y,\xi_0,\dots,\xi_N]$. Then these polynomials define  the
affine variety $X\subset\A_F^{N+3}$. For $H\in
F[x,y,\xi_0,\dots,\xi_N]$, denote by $D(H)$ the open complement
of the variety defined by $H$ in $\A_F^{N+3}$. Set
$$
\Symb(\cA)=X\cap D(x)\cap D(y) \cap \left(\cup_{i=0}^N
D(\xi_i)\right).
$$
Now we will show that $\Symb(\cA)$ has the required properties.
Assume $(x_0,y_0,c_0,\dots,c_N)\in \Symb(\cA)(L)$ for a field
extension $L/F$. Hence
\begin{equation} \label{eq:polynoms}
F_j(x_0,y_0,c_0,\dots,c_N)=0
\end{equation} for any
$j\in J$. Note that the specialization $x\mapsto x_0$, $y\mapsto
y_0$ of polynomials $F_j\in F[x,y,\xi_0,\dots,\xi_N]$ gives the
polynomials defining the Severi-Brauer variety of the algebra
$(x_0,y_0)_n\otimes\cA^{op}_L$. The condition (\ref{eq:polynoms})
shows that $\SB((x_0,y_0)_n\otimes\cA_L^{op})$ has an $L$-rational
point. Then $(x_0,y_0)_n\otimes\cA^{op}_L$ is a matrix algebra,
that is $(x_0,y_0)_n$ is Brauer equivalent to $\cA_L$. Since $\deg
(\cA_L)= \deg ((x_0,y_0)_n)$, we have $(x_0,y_0)\cong \cA_L$.

Now assume that $(x_0,y_0)_n\cong \cA_L$ for some $x_0,y_0\in L^*$.
Then 
$$
\SB((x_0,y_0)_n\otimes\cA_L^{op})
$$ 
has an $L$-rational point,
say, $(c_0,\dots,c_N)$. That is, $F_j(x_0,y_0,c_0,\dots,c_N)=0$ for
any $j\in J$. Hence $(x_0,y_0,c_0,\dots,c_N)\in X$. Since
$x_0,y_0\in L^*$ and some of $c_i$-s is not zero, then
$(x_0,y_0,c_0,\dots,c_N)\in \Symb(\cA)$.

 \qed

Finally, one can easy prove the following property of $\Symb(\cA)$.

\begin{prop} \label{p:sym var extension}
Let $\cA/F$ be a central simple algebra. Then, for any field
extension $K/F$,
$$
\Symb(\cA\otimes_F K)= \Symb(\cA)\times_F K.
$$
\end{prop}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{The variety $C(\cA, K)$} \label{sec:cycl var}



 Let $\cA$ be a central simple algebra over a field $F$ of
degree $n$ and $K/F$ a cyclic field extension of degree $n$.


Consider a cyclic $F(x)$-algebra
$$
\cC=(K(x)/F(x),\sigma, x),
$$
where $\Gal(K(x)/F(x))=\< \sigma \> $. One has $\cC=\sum_{0\le i\le
n-1}u_{\sigma^i}K(x)$ and
$$
u_{\sigma^i} u_{\sigma^j}=
 \left\{
\begin{array}{lc}
u_{\sigma^{i+j}}& \mbox{if } i+j<n, \\
x u_{\sigma^{i+j-n}} &	\mbox{if } i+j\ge n. \\
\end{array}
\right.
$$

Let $b_1, \dots, b_n$ be a base of an $F$-vector space $K$ and $v_1,
\dots, v_{n^2}$ a base of a central simple $F$-algebra $\cB$ of
degree $n$ consisting of invertible elements.

Then the multiplicative group generated by the set
$$
\{u_{\sigma},b_1, \dots, b_n, v_1, \dots, v_{n^2}\}
$$
contains a base of
$$
\cD=\cC\otimes_{F(x)} \cB_{F(x)}.
$$
Then, as in the section \ref{sec:symbol var}, we can construct
polynomials 
$$
F_j\in F(x)[\xi_0,\dots,\xi_N], 
$$
$j\in J$, defining
the Severi-Brauer variety $\SB(\cD)$, where $N=C_{n^4}^{n^2}-1$.
Moreover, one can prove that these polynomials have coefficients
not only in $F(x)$, but in $F[x]$.

Let $L/F$ be a field extension such that $[L K:L]=[K:F]$. For
$a\in L^*$, consider the algebra
$$
\overline{\cD}=(L K/L,\tau, a)\otimes_L \cB_L,
$$
where $\Gal(L K/L)=\<\tau\>.$ One has $(L K/L,\tau, a)=\sum_{0\le i\le
n-1}w_{\tau^i}L K$ and
$$
w_{\tau^i} w_{\tau^j}=
 \left\{
\begin{array}{lc}
w_{\tau^{i+j}}& \mbox{if } i+j<n, \\
a w_{\tau^{i+j-n}} &  \mbox{if } i+j\ge n. \\
\end{array}
\right.
$$

Since $[L K:L]=[K:F]$, then $b_1, \dots, b_n$ is a base of an
$L$-vector space $L K$. Then the multiplicative group generated
by the set
$$
\{w_{\tau},b_1, \dots, b_n,v_1, \dots, v_{n^2} \}
$$
contains a base of $\overline{\cD}$.

Proceeding as in the proof of Lemma \ref{l:spec1} we can prove

\begin{lemma}  \label{l:spec2}
Let $\cB$ be a central simple $F$-algebra of degree $n$, $K/F$ a
cyclic field extension of degree $n$.  Let also
$$
\cD=(K(x)/F(x),\sigma, x)\otimes_{F(x)} \cB_{F(x)}.
$$
Then the Severi-Brauer variety $\SB(\cD)$ of $\cD$ can be defined
by polynomials in $F[x][\xi_0,\dots,\xi_N]$ such that 
for any field extension $L/F$ and $a\in L^*$
their
specialization $x\mapsto a$ gives polynomials in
$L[\xi_0,\dots,\xi_N]$ defining the Severi-Brauer variety
$\SB(\overline{\cD})$ of $\overline{\cD}=(L K/L,\tau, a)\otimes_L \cB_L$.
\end{lemma}

Now we can construct the variety $C(\cA,K)$.

%\begin{theorem}  \label{t:main2}
%Let $\cA$ be a central simple algebra over a field $F$ of degree
%$n$ and $K/F$ a cyclic field extension of degree $n$.	We
%construct a quasi-affine variety $C(\cA, K)$ such that for a
%field extension $L/F$ with the property $[K L:L]=[K:F]$ $C(\cA,
%K)$ has an $L$-rational point iff $K L$ is a subfield of $\cA_L$.
%\end{theorem}

\noindent {\it Proof  of Theorem \ref{t:main cycle var}}. Set
$$
\cD=(K(x)/F(x),\sigma, x)\otimes_{F(x)} \cA_{F(x)},
$$
where $F(x)$ is a purely transcendental extension of $F$.

The Severi-Brauer variety of $\cD$ is a closed subvariety of the
projective space $\PP_{F(x)}^N$, where $N=C_{n^4}^{n^2}-1$. Let
$F_j\in F[x][\xi_0,\dots,\xi_N]$, $j\in J$, be polynomials defining
$\SB(\cD)$ constructed in the proof of Lemma \ref{l:spec2}. Now
consider $F_j$ as polynomials in $F[x,\xi_0,\dots,\xi_N]$. Then
these polynomials define an affine variety $X\subset\A_F^{N+2}$.
For $H\in F[x,\xi_0,\dots,\xi_N]$, denote by $D(H)$ the open
complement of the variety defined by $H$ in $\A_F^{N+2}$. Set
$$
C(\cA,K)=X\cap D(x) \cap \left(\cup_{i=0}^N D(\xi_i)\right).
$$

The rest of the proof is the same as in Theorem \ref{t:main}.

\qed

As in the case of  $\Symb(\cA)$, we have the following property of
$C(\cA,K)$.

\begin{prop} \label{p:sym var extension}

Let $\cA$ be a central simple algebra over a field $F$ of degree
$n$ and $K/F$ a cyclic field extension of degree $n$. Let also
$L/F$ be a field extension such that $|K:F| = |LK:L|$.

Then
$$
C(\cA\otimes_F L, LK)= C(\cA,K)\times_F L.
$$
\end{prop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}
