An Algorithmic Approach to Invariant Rational Functions

Abstract

For any field K and group A acting on K(x0, x1,..., xn-1), the fixed field consists of the elements fixed under the action of A, and is denoted K(x0, x1,..., xn-1)^A. Noether's problem seeks to answer whether K(x0, x1,..., xn-1)^A/K is a purely transcendental field extension. Lenstra gave necessary and sufficient conditions for a purely transcendental extension when A is any finite abelian group, however his proof is often regarded as non-constructive. Masuda constructed generators of the fixed field when a certain ideal is principal in an integral group ring, and exhibited results when A is a cyclic group of order n <= 7 and n=11. We prove, however, that the ideal in question is not principal when n=13, 17, 19 or 23, and thus Masuda's techniques cannot be used. We use Lenstra's proof as a basis of an algorithm which computes generators of the fixed field. We demonstrate this algorithm for groups of order n=3, 5, 7, 11, 13, 17, 19 and 23. Swan gave the first negative answer to Noether's problem for the cyclic group of order 47 over Q. Leshin quantifies the degree to which a field extension fails to be purely transcendental by defining the degree of irrationality. We use the fact that the class group of the maximal real subfield Q(zeta_23+zeta_23^-1) is trivial to construct a field extension of degree 47 below the fixed field that is purely transcendental over K, thus providing a new bound for the degree of irrationality.