  This paper is dedicated to the conjugacy problem in $GL(n,\Z)$ and its
connection with algebraic number theory. This connection may be summed up in
the Latimer-MacDuffee-Taussky Theorem, which, in a very broad sense, identifies
the conjugacy relation in $GL(n,\Z)$ with the relation of arithmetic
equivalence between certain ideals of algebraic numbers. The main purpouse is
to clarify the link between a weaker relation between ideals, weak equivalence,
and local conjugacy in $GL(n,\Z)$, i.e., the conjugacy of $GL(n,\Z)$ matrices
in the groups $GL(n,\Z_{p})$ of invertible matrices over the $p$-adic integers.
