theorems on continuation

Theorem 1.  When ν0 is an exponent valuation of the field k and K/k is a finite field extension, ν0 has a continuation to the extension fieldMathworldPlanetmath K.

Theorem 2.  If the degree ( of the field extension K/k is n and ν0 is an arbitrary exponent ( of k, then ν0 has at most n continuations to the extension field K.

Theorem 3.  Let ν0 be an exponent valuation of the field k and 𝔬 the ring of the exponent ν0.  Let K/k be a finite extension and 𝔒 the integral closureMathworldPlanetmath of 𝔬 in K.  If  ν1,,νm are all different continuations of ν0 to the field K and 𝔒1,,𝔒m their rings (, then


The proofs of those theorems are found in [1], which is available also in Russian (original), English and French.

Corollary.  The ring 𝔒 (of theorem 3) is a UFD.  The exponents of K, which are determined by the pairwise coprime prime elementsMathworldPlanetmath of 𝔒, coincide with the continuations ν1,,νm of ν0.  If π1,,πm are the pairwise coprime prime elements of 𝔒 such that  νi(π1)=1  for all  i’s and if the prime element p of the ring 𝔬 has the


with ε a unit of 𝔒, then ei is the ramification index of the exponent νi with respect to ν0 (i=1,,m).


  • 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title theorems on continuation
Canonical name TheoremsOnContinuation
Date of creation 2013-03-22 17:59:51
Last modified on 2013-03-22 17:59:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 12J20
Classification msc 13A18
Classification msc 13F30
Classification msc 11R99
Synonym theorems on continuations of exponents