# divisor theory in finite extension

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 (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 $\mathcal{O}$ 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 $\mathbb{Q}(\vartheta)$.

## References

• 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title divisor theory in finite extension DivisorTheoryInFiniteExtension 2013-03-22 17:59:59 2013-03-22 17:59:59 pahio (2872) pahio (2872) 7 pahio (2872) Theorem msc 13A18 msc 13F05 msc 12J20 msc 13A05 msc 11A51 FiniteExtensionsOfDedekindDomainsAreDedekind