ring of exponent

Definition.  Let ν be an exponent valuation of the field K.  The subring


of K is called the ν.  It is, naturally, an integral domainMathworldPlanetmath.  Its elements are called ν.

Theorem 1.  The ring of the exponent ν of the field K is integrally closedMathworldPlanetmath in K.

Theorem 2.  The ring 𝒪ν only one prime elementMathworldPlanetmath π, when one does not regard associated elements as different.  Any non-zero element α can be represented uniquely with a π in the form


where ε is a unit of 𝒪ν and  m=ν(α)0.  This means that 𝒪 is a UFD.

Remark 1.  The prime elements π of the ring 𝒪ν are characterised by the equation  ν(π)=1  and the units  ε the equation  ν(ε)=0.

Remark 2.  In an algebraically closed field Ω, there are no exponents (http://planetmath.org/ExponentValuation).  In fact, if there were an exponent ν of Ω and if π were a prime element of the ring of the exponent, then, since the equation  x2-π=0  would have a root (http://planetmath.org/Equation) ϱ in Ω, we would obtain  2ν(ϱ)=ν(ϱ2)=ν(π)=1;  this is however impossible, because an exponent attains only integer values.

Theorem 3.  Let  𝔒1,,𝔒r be the rings of the different exponent valuations ν1,,νr of the field K.  Then also the intersection


is a subring of K with unique factorisation (http://planetmath.org/UFD).  To be precise, any non-zero element α of 𝔒 may be uniquely represented in the form


in which ε is a unit of 𝔒,  the integers n1,,nr are nonnegative and π1,,πr are coprimeMathworldPlanetmathPlanetmath prime elements of 𝔒 satisfying

νi(πj)=δij={1for i=j,0for ij.
Title ring of exponent
Canonical name RingOfExponent
Date of creation 2013-03-22 17:59:43
Last modified on 2013-03-22 17:59:43
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Definition
Classification msc 13F30
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Related topic DiscreteValuationRing
Related topic ValuationRingOfAField
Related topic LocalRing
Defines ring of an exponent
Defines ring of the exponent
Defines integral with respect to an exponent