  Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two
number fields. Assume that there is a group isomorphism of points on $G$ over
the finite 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. Under these conditions, if $K$ or $L$ is a
Galois extension of $\mathbf{Q}$ and $G(\mathbf{A}_{K,f})$ and
$G(\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as
fields. We use this result to show that if for two number fields $K$ and $L$
that are Galois over $\mathbf{Q}$, the finite Hecke algebras for
$\mathrm{GL}(n)$ (for fixed $n > 1$) are isomorphic by an isometry for the
$L^1$-norm, then 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 $\mathbf{Q}$.
