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 $\nu_0$ ($i = 1,\,\ldots,\,m$ ).

Bibliography

1

S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).