  We construct examples of number fields which are not isomorphic but for which
their idele class groups are isomorphic. We also construct examples of
projective algebraic curves which are not isomorphic but for which their
Jacobian varieties are isomorphic. Both are constructed using an example in
group theory provided by Leonard Scott of a finite group $G$ and subgroups
$H_1$ and $H_2$ which are not conjugate in $G$ but for which the $G$-module
${\mathbb Z}[G/H_1]$ is isomorphic to ${\mathbb Z}[G/H_2]$.
