The q-division ring and its fixed rings

We prove that the fixed ring of the q-division ring k(q)(x, y) under any finite group of monomial automorphisms is isomorphic to k(q)(x,y) for the same q. In a similar manner, we also show that this phenomenon extends to an automorphism that is defined only on k(q)(x,y) and does not restrict to k(q)[x(+/- 1), y(+/- l)]. We then use these results to answer several questions posed by Artamonov and Cohn about the endomorphisms and automorphisms of k(q)(x,y). (C) 2014 Elsevier Inc. All rights reserved.