in 1918, noether published a paper on the inverse galois problem. instead of determining the galois group of transformations of a given field and its extension, noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its galois group. she reduced this to "noether's problem", which asks whether the fixed field of a subgroup "g" of the permutation group "s" acting on the field "k"("x", ... , "x") always is a pure transcendental extension of the field "k". (she first mentioned this problem in a 1913 paper, where she attributed the problem to her colleague fischer.) she showed this was true for "n" = 2, 3, or 4. in 1969, r.g. swan found a counter-example to noether's problem, with "n" = 47 and "g" a cyclic group of order 47 (although this group can be realized as a galois group over the rationals in other ways). the inverse galois problem remains unsolved.