divisor theory in finite extension

Theorem.  Let the integral domainMathworldPlanetmath π’ͺ, with the quotient field k, have the divisor theory  π’ͺ*→𝔇, determined (see divisors and exponents) by the exponent (http://planetmath.org/ExponentValuation2) system N0 of k.  If K/k is a finite extensionMathworldPlanetmath, then the exponent system N, consisting of the continuations (http://planetmath.org/ContinuationOfExponent) of all exponents in N0 to the field K, determines the divisor theory of the integral closureMathworldPlanetmath of π’ͺ in K.

Corollary.  In the ring of integers π’ͺ of any algebraic number fieldMathworldPlanetmath β„šβ’(Ο‘), there is a divisor theory π’ͺ*→𝔇, determined by the set of all exponent valuations of β„šβ’(Ο‘).


  • 1 S. Borewicz & I. Safarevic: Zahlentheorie.  BirkhΓ€user Verlag. Basel und Stuttgart (1966).
Title divisor theory in finite extension
Canonical name DivisorTheoryInFiniteExtension
Date of creation 2013-03-22 17:59:59
Last modified on 2013-03-22 17:59:59
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 13A18
Classification msc 13F05
Classification msc 12J20
Classification msc 13A05
Classification msc 11A51
Related topic FiniteExtensionsOfDedekindDomainsAreDedekind