  We show that the presentation of affine $\mathbb{T}$-varieties of complexity
one in terms of polyhedral divisors holds over an arbitrary field. We also
describe a class of multigraded algebras over Dedekind domains. We study how
the algebra associated to a polyhedral divisor changes when we extend the
scalars. As another application, we provide a combinatorial description of
affine $\mathbf{G}$-varieties of complexity one over a field, where
$\mathbf{G}$ is a (not-nescessary split) torus, by using elementary facts on
Galois descent. This class of affine $\mathbf{G}$-varieties is described via a
new combinatorial object, which we call (Galois) invariant polyhedral divisor.
