# 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 $\U0001d52c$ the ring of the exponent ${\nu}_{0}$. Let $K/k$ be a finite extension and $\U0001d512$ the integral closure^{} of $\U0001d52c$ in $K$. If ${\nu}_{1},\mathrm{\dots},{\nu}_{m}$ are all different continuations of ${\nu}_{0}$ to the field $K$ and ${\U0001d512}_{1},\mathrm{\dots},{\U0001d512}_{m}$ their rings (http://planetmath.org/RingOfExponent), then

$$\U0001d512=\bigcap _{i=1}^{m}{\U0001d512}_{i}.$$ |

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

Corollary. The ring $\U0001d512$ (of theorem 3) is a UFD. The exponents of $K$, which are determined by the pairwise coprime prime elements^{} of $\U0001d512$, 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 $\U0001d512$ such that ${\nu}_{i}({\pi}_{1})=1$ for all $i$’s and if the prime element $p$ of the ring $\U0001d52c$ has the

$$p=\epsilon {\pi}_{1}^{{e}_{1}}\mathrm{\cdots}{\pi}_{m}^{{e}_{m}}$$ |

with $\epsilon $ a unit of $\U0001d512$, then ${e}_{i}$ is the ramification index of the exponent ${\nu}_{i}$ with respect to ${\nu}_{0}$ ($i=1,\mathrm{\dots},m$).

## References

- 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).

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 |