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.