let g denote an algebraic group over q and k and l two number fields. assume that there is a group isomorphism of points on g over the adeles of k and l, respectively. we establish conditions on the group g, related to the structure and the splitting field of its borel groups, under which k and l have isomorphic adele rings. under these conditions, if k or l is a galois extension of q and $g(a_k)$ and $g(a_l)$ are isomorphic, then k and l are isomorphic as fields. as a corollary, we show that an isomorphism of hecke algebras for $gl(n)$ (for fixed $n > 1$), which is an isometry in the $l^1$ norm over two number fields k and l that are galois over q, implies that the fields k and l are isomorphic. this can be viewed as an analogue in the theory of automorphic representations of the theorem of neukirch that the absolute galois group of a number field determines the field if it is galois over q.