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Hometheorems on continuation

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# 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 of the field extension $K/k$ is $n$ and $\nu_{0}$ is an arbitrary exponent 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, 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 representation

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

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).

## Mathematics Subject Classification

12J20*no label found*13A18

*no label found*13F30

*no label found*11R99

*no label found*

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