# divisor theory in finite extension

Theorem.β Let the integral domain^{} $\mathrm{\pi \x9d\x92\u037a}$, with the quotient field $k$, have the divisor theoryβ ${\mathrm{\pi \x9d\x92\u037a}}^{*}\beta \x86\x92\mathrm{\pi \x9d\x94\x87}$, determined (see divisors and exponents) by the exponent (http://planetmath.org/ExponentValuation2) system ${N}_{0}$ of $k$.β If $K/k$ is a finite extension^{}, then the exponent system $N$, consisting of the continuations (http://planetmath.org/ContinuationOfExponent) of all exponents in ${N}_{0}$ to the field $K$, determines the divisor theory of the integral closure^{} of $\mathrm{\pi \x9d\x92\u037a}$ in $K$.

Corollary.β In the ring of integers $\mathrm{\pi \x9d\x92\u037a}$ of any algebraic number field^{} $\mathrm{\beta \x84\x9a}\beta \x81\u2019(\mathrm{{\rm O}\x91})$, there is a divisor theory ${\mathrm{\pi \x9d\x92\u037a}}^{*}\beta \x86\x92\mathrm{\pi \x9d\x94\x87}$, determined by the set of all exponent valuations of $\mathrm{\beta \x84\x9a}\beta \x81\u2019(\mathrm{{\rm O}\x91})$.

## References

- 1 S. Borewicz & I. Safarevic: Zahlentheorie.β BirkhΓ€user Verlag. Basel und Stuttgart (1966).

Title | divisor theory in finite extension |
---|---|

Canonical name | DivisorTheoryInFiniteExtension |

Date of creation | 2013-03-22 17:59:59 |

Last modified on | 2013-03-22 17:59:59 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13A18 |

Classification | msc 13F05 |

Classification | msc 12J20 |

Classification | msc 13A05 |

Classification | msc 11A51 |

Related topic | FiniteExtensionsOfDedekindDomainsAreDedekind |