First, let $K$ and $L$ be number fields and let $G$ be a linear algebraic group over ${\bf Q}$. Suppose that there is a topological 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 of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. As a corollary, we show that when $K$ and $L$ are Galois over ${\bf Q}$, an isomorphism of Hecke algebras for $GL(n)$, which is an isometry in the $L_1$-norm, implies that $K$ and $L$ are isomorphic as fields. 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 ${\bf Q}$.
Secondly, let $K$ and $L$ be non-archimedean local fields of characteristic zero. Analogous results to the number fields case still hold. However, we show that the Hecke algebra for $GL(2)$ for any local field is Morita equivalent to the same complex algebra, determined by the Bernstein decomposition.