eq
(L*\mathcal M)^{G}$ is an Ore domain, $S=\Gamma\setminus \{0\}$. Hence for any $u_{1},u_{2}\in U$ there exist $v_{1},v_{2}\in U,s_{1},s_{2}\in S$, such that $u_{1}v_{1}s_{1}^{-1}=u_{2}v_{2}s_{2}^{-1}$. Since $S$ is commutative we get $u_{1}v_{1}s_{2}=u_{2}v_{2}s_{1}$. Other conditions of Ore rings are proved analogously.

Let $V$ be a $\c$-vector space, $\dim_{\c}V=N$, $L$ the field of fractions of the symmetric algebra $S(V)$ and $G\subset {{\rm{Aut}_{\c}}}(L)$ is the classical reflection group whose action on $L$ is induced by its action by reflections on $V$. Then $L=\c(t_{1},\dots, t_{N})$ and $K=L^G$ is a purely transcendental extension of $\c$ by the Chevalley-Shephard-Todd Theorem. If $\mathcal M\subset {{\rm{Aut}_{\c}}}(L)$ is a subgroup such that $G$ normalizes $\mathcal M$ and $U$ is a Galois algebra in $(L*\mathcal M)^G$ with respect to a polynomial subalgebra $\Gamma$ such that the field of fractions $F(\Gamma)$ is isomorphic to $ K$, then such $U$ will be called a *linear Galois algebra*. In particular, $\mathcal K=(L*\mathcal M)^G$ is itself a linear Galois algebra with respect to $\Gamma$ if $L$, $G$, $\mathcal M$, $K$ and $\Gamma$ are as above. Other examples of linear Galois algebras are the universal enveloping algebras of $gl_{n}$ and $sl_{n}$ with respect to their Gelfand-Tsetlin subalgebras.

We will need the following action of $\mathcal M$ on $L$. Suppose $\mathcal M = \mathbb Z^{N}$ is a free abelian group of rank $N$ generated by $\sigma_{i}, i=1, \ldots, N$. We say that $\mathbb Z^{N}$ acts by shifts on $L$ if $\sigma_i$, $i=1, \ldots, N$ act on $t_j$ as follows: $\sigma_i(t_j)=t_j-\delta_{ij}$. Thus we can construct a skew group algebra $L*\mathcal M$. As a subgroup of automorphisms of $L$, $G$ normalizes $\mathcal M$ $$\sigma_{k}^{\pi}=\delta_{\pi(k)}, \pi\in S_{p}, \sigma_{k}^{\epsilon_{i}}=\left\{\begin{array}
{ll}\sigma_{k},&\text{ if } i\neq k\\
\sigma_{k}^{-1}&\text{ otherwise, }
\end{array}\right.$$

Hence $G$ acts on $L*\mathcal M$. We will call this action *natural*.

Let $R