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 $\mathfrak{o}$ the ring of the exponent $\nu_{0}$.  Let $K/k$ be a finite extension and $\mathfrak{O}$ the integral closure of $\mathfrak{o}$ in $K$.  If  $\nu_{1},\,\ldots,\,\nu_{m}$ are all different continuations of $\nu_{0}$ to the field $K$ and $\mathfrak{O}_{1},\,\ldots,\,\mathfrak{O}_{m}$ their rings (http://planetmath.org/RingOfExponent), then

 $\mathfrak{O}=\bigcap_{i=1}^{m}\mathfrak{O}_{i}.$

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

Corollary.  The ring $\mathfrak{O}$ (of theorem 3) is a UFD.  The exponents of $K$, which are determined by the pairwise coprime prime elements of $\mathfrak{O}$, coincide with the continuations $\nu_{1},\,\ldots,\,\nu_{m}$ of $\nu_{0}$.  If $\pi_{1},\,\ldots,\,\pi_{m}$ are the pairwise coprime prime elements of $\mathfrak{O}$ such that  $\nu_{i}(\pi_{1})=1$  for all  $i$’s and if the prime element $p$ of the ring $\mathfrak{o}$ has the

 $p=\varepsilon\pi_{1}^{e_{1}}\cdots\pi_{m}^{e_{m}}$

with $\varepsilon$ a unit of $\mathfrak{O}$, then $e_{i}$ is the ramification index of the exponent $\nu_{i}$ with respect to $\nu_{0}$ ($i=1,\,\ldots,\,m$).

References

• 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title theorems on continuation TheoremsOnContinuation 2013-03-22 17:59:51 2013-03-22 17:59:51 pahio (2872) pahio (2872) 10 pahio (2872) Theorem msc 12J20 msc 13A18 msc 13F30 msc 11R99 theorems on continuations of exponents