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 (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 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 ${\nu }_0$ ($i=1,\mathrm{\dots },m$).