# Rationality Problems for Multiplicative Invariant Fields

Nicole Lemire

(Waterloo U.)

\begin{abstract}

For a finite group $G$, a $\z G$ lattice $A$ and a field $K$ on which $G$ acts trivially, the action of the group $G$ can be extended naturally to an action on the quotient field $K(A)$ of the group algebra $K[A]$. The fixed field $K(A)^G$ of $K(A)$ under this action of $G$ is called a multiplicative invariant field. We prove that the rationality problem for a given multiplicative invariant field $K(A)^G$ over $K(A^R)^G$ where $R$ is the normal subgroup of $G$ generated by reflections acting on the lattice $A$ is equivalent to the rationality problem for the multiplicative invariant field $K(A)^{\Omega_G}$ over $K(A^R)^{\Omega_G}$, where $\Omega_G$ is a particular subgroup of $G$ satisfying $G/R\cong \Omega_G$. We use this result to prove that $K(A)^G$ is rational over $K$ where $G$ is the automorphism group of a crystallographic root system $\Psi$ acting on $V= \q\Psi$ and $A$ is any $\z G$ lattice on $V$. \end{abstract}