divisors in base field and finite extension field

Let k be the quotient field of an integral domainMathworldPlanetmath 𝔬 which has the divisor theoryMathworldPlanetmath  𝔬*→𝔇0.  Let K/k a finite extensionMathworldPlanetmath, 𝔒 be the integral closureMathworldPlanetmath of 𝔬 in K and  𝔒*→𝔇  the uniquely determined divisor theory of 𝔒 (see the parent entry (http://planetmath.org/DivisorTheoryInFiniteExtension)).  We will study the of the divisorMathworldPlanetmathPlanetmath monoids 𝔇0 and 𝔇.

Any element a of 𝔬*, which is a part of 𝔒*, determines a principal divisor  (a)kβˆˆπ”‡0  and another  (a)Kβˆˆπ”‡.  The (multiplicative) monoid 𝔬* is isomorphically embedded (via ΞΉ) in the monoid 𝔒*.  Because the units of the ring 𝔒, which belong to 𝔬, are all units of 𝔬 and because associatesMathworldPlanetmath always determine the same principal divisor, the mentioned embedding defines an isomorphicPlanetmathPlanetmathPlanetmath mapping

(a)k↦(a)K (1)

from the monoid of the principal divisors of 𝔬 into the monoid of the principal divisors of 𝔒.  One has the

Theorem.  There is one and only one isomorphismMathworldPlanetmathPlanetmath Ο† from the divisor monoid 𝔇0 into the divisor monoid 𝔇 such that its restriction to the principal divisors of 𝔬 coincides with (1).  Then there is the following commutative diagramMathworldPlanetmath:


The isomorphism  Ο†:𝔇o→𝔇  is determined as follows.  Let 𝔭 be an arbitrary prime divisor in 𝔇0 and ν𝔭 the corresponding exponent valuation of the field k.  Let  ν𝔓1,…,ν𝔓m  be the continuations of the exponentPlanetmathPlanetmath ν𝔭 to K, which correspond to the prime divisors  𝔓1,…,𝔓m in 𝔇.  If  e1,…,em  are the ramification indices of the exponents  ν𝔓1,…,ν𝔓m  with respect to ν𝔭, then we have


Thus apparently, the factor of the principal divisor  (a)Kβˆˆπ”‡,  which corresponds to the factor 𝔭ν𝔭⁒(a) of the principal divisor  (a)kβˆˆπ”‡0, is  (𝔓1e1⁒⋯⁒𝔓mem)ν𝔭⁒(a).  Then Ο† is settled by


When one identifies 𝔇0 with its isomorphic image φ⁒(𝔇0), we can write


i.e. the prime divisors in 𝔇0 don’t in general remain as prime divisors in 𝔇.  On grounds of the identification one may speak of the divisibility of the divisors of 𝔬 by the divisors of 𝔒.  The coprimeMathworldPlanetmathPlanetmath divisors of 𝔬 are coprime also as divisors of 𝔒.


  • 1 S. Borewicz & I. Safarevic: Zahlentheorie.  BirkhΓ€user Verlag. Basel und Stuttgart (1966).
Title divisors in base field and finite extension field
Canonical name DivisorsInBaseFieldAndFiniteExtensionField
Date of creation 2013-03-22 18:01:35
Last modified on 2013-03-22 18:01:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Topic
Classification msc 13F05
Classification msc 13A18
Classification msc 13A05
Related topic DivisorTheory