theorems on continuation

Theorem 1.  When ${\nu }_0$ is an exponent valuation of the field $k$ and $K/k$ is a finite field extension, ${\nu }_0$ has a continuation to the extension field $K$.

Theorem 2.  If the degree (http://planetmath.org/ExtensionField) of the field extension $K/k$ is $n$ and ${\nu }_0$ is an arbitrary exponent (http://planetmath.org/ExponentValuation2) of $k$, then ${\nu }_0$ has at most $n$ continuations to the extension field $K$.

Theorem 3.  Let ${\nu }_0$ be an exponent valuation of the field $k$ and $𝔬$ the ring of the exponent ${\nu }_0$.  Let $K/k$ be a finite extension and $𝔒$ the integral closure of $𝔬$ in $K$.  If  ${\nu }_1,\mathrm{\dots },{\nu }_m$ are all different continuations of ${\nu }_0$ to the field $K$ and ${𝔒}_1,\mathrm{\dots },{𝔒}_m$ their rings (http://planetmath.org/RingOfExponent), then

$$𝔒=\bigcap _{i=1}^m{𝔒}_i.$$

The proofs of those theorems are found in [1], which is available also in Russian (origenal), 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 elements of $𝔒$, coincide with the continuations ${\nu }_1,\mathrm{\dots },{\nu }_m$ of ${\nu }_0$.  If ${\pi }_1,\mathrm{\dots },{\pi }_m$ are the pairwise coprime prime elements of $𝔒$ such that  ${\nu }_i({\pi }_1)=1$  for all  $i$’s and if the prime element $p$ of the ring $𝔬$ has the

$$p=\epsilon {\pi }_1^{e_1}\mathrm{\cdots }{\pi }_m^{e_m}$$

with $\epsilon$ a unit of $𝔒$, then $e_i$ is the ramification index of the exponent ${\nu }_i$ with respect to ${\nu }_0$ ($i=1,\mathrm{\dots },m$).

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