continuation of exponent

Theorem.  Let K/k be a finite field extension and ν an exponent valuation of the extension fieldMathworldPlanetmath K.  Then there exists one and only one positive integer e such that the function

(1)      ν0(x):={when x=0,ν(x)ewhen x0,

defined in the base field k, is an exponent ( of k.

Proof.  The exponent ν of K attains in the set k{0} also non-zero values; otherwise  k would be included in 𝒪ν, the ring of the exponent ν.  Since any element ξ of K are integral over k, it would then be also integral over 𝒪ν, which is integrally closed in its quotient field K (see theorem 1 in ring of exponent); the situation would mean that  ξ𝒪ν and thus the whole K would be contained in 𝒪ν.  This is impossible, because an exponent of K attains also negative values.  So we infer that ν does not vanish in the whole k{0}.  Furthermore, ν attains in k{0} both negative and positive values, since  ν(a)+ν(a-1)=ν(aa-1)=ν(1)=0.

Let p be such an element of k on which ν attains as its value the least possible positive integer e in the field k and let a be an arbitrary non-zero element of k.  If

ν(a)=m=qe+r(q,r,  0r<e),

then  ν(ap-q)=m-qe=r,  and thus  r=0  on grounds of the choice of p.  This means that ν(a) is always divisible by e, i.e. that the values of the function ν0 in k{0} are integers.  Because  ν0(p)=1  and  ν0(pl)=l,  the function attains in k every integer value.  Also the conditions


are in , whence ν0 is an exponent of the field k.

Definition.  Let K/k be a finite field extension.  If the exponent ν0 of k is tied with the exponent ν of K via the condition (1), one says that ν induces ν0 to k and that ν is the continuation of ν0 to K.  The positive integer e, uniquely determined by (1), is the ramification index of ν with respect to ν0 (or with respect to the subfieldMathworldPlanetmath k).


  • 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title continuation of exponent
Canonical name ContinuationOfExponent
Date of creation 2013-03-22 17:59:49
Last modified on 2013-03-22 17:59:49
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Definition
Classification msc 13A18
Classification msc 12J20
Classification msc 11R99
Classification msc 13F30
Synonym prolongation of exponent
Defines induce
Defines continuation
Defines continuation of the exponent
Defines ramification index of the exponent