divisor theory in finite extension

Theorem.  Let the integral domain $𝒪$, with the quotient field $k$, have the divisor theory  ${𝒪}^{*}\to 𝔇$, determined (see divisors and exponents) by the exponent (http://planetmath.org/ExponentValuation2) system $N_0$ of $k$.  If $K/k$ is a finite extension, then the exponent system $N$, consisting of the continuations (http://planetmath.org/ContinuationOfExponent) of all exponents in $N_0$ to the field $K$, determines the divisor theory of the integral closure of $𝒪$ in $K$.

Corollary.  In the ring of integers $𝒪$ of any algebraic number field $\mathbb{Q}(\vartheta )$, there is a divisor theory ${𝒪}^{*}\to 𝔇$, determined by the set of all exponent valuations of $\mathbb{Q}(\vartheta )$.