  It is shown that the q-difference Noether problem for all classical Weyl
groups has a positive solution, simultaneously generalizing well known results
on multisymmetric functions of Mattuck and Miyata in the case q=1, and
q-deforming the noncommutative Noether problem for the symmetric group. It is
also shown that the quantum Gelfand-Kirillov conjecture for gl_N (for a generic
q) follows from the positive solution of the q-difference Noether problem for
the Weyl group of type D_n. The proof is based on the theory of Galois rings
developed by the first author and Ovsienko. From here we obtain a new proof of
the quantum Gelfand-Kirillov conjecture for sl_N, thus recovering the result of
Fauquant-Millet. Moreover, we provide an explicit description of skew fields of
fractions for quantized gl_N and sl_N generalizing Alev and Dumas.
