  The theory of factor-equivalence of integral lattices gives a far reaching
relationship between the Galois module structure of units of the ring of
integers of a number field and its arithmetic. For a number field $K$ that is
Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary
and sufficient condition on the quotients of class numbers of subfields of $K$,
for the quotient $E_{K}$ of the group of units of the ring of integers of $K$
by the subgroup of roots of unity to be factor equivalent to the standard
cyclic Galois module. By using strong arithmetic properties of totally real
$p$-rational number fields, we prove that the non-abelian $p$-rational
$p$-extensions of $\mathbb{Q}$ do not have Minkowski units, which extends a
result of Burns to non-abelian number fields. We also study the relative Galois
module structure of $E_{L}$ for varying Galois extensions $L/F$ of totally real
$p$-rational number fields whose Galois groups are isomorphic to a fixed finite
group $G$. In that case, we prove that there is a finite set $\Omega$ of
$\mathbb{Z}_p[G]$-lattices such that for every $L$, $\mathbb{Z}_{p}
\otimes_{\mathbb{Z}} E_{L}$ is factor equivalent to $\mathbb{Z}_{p}[G]^{n}
\oplus X$ as $\mathbb{Z}_p[G]$-lattices for some $X \in \Omega$ and an integer
$n \geq 0$.
