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

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

Bibliography

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S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).