# Riemann-Hurwitz theorem

First we define the different divisor^{} of an extension of function fields. Let $K$ be a function field^{} over a field $F$ and let $L$ be a finite separable extension^{} of $K$. Let ${\mathcal{O}}_{P}$ be a prime of $K$, i.e. a discrete valuation ring with $F\subset {\mathcal{O}}_{P}$, maximal ideal^{} $P$ and quotient field equal to $K$. Let ${R}_{P}$ be the integral closure^{} of ${\mathcal{O}}_{P}$ in $L$. Notice that if $\U0001d52d$ is a prime ideal^{} of ${R}_{P}$, then the localization^{} ${\mathcal{O}}_{\U0001d52d}={({R}_{P})}_{\U0001d52d}$ is a prime of $L$ (which is said to be lying over ${\mathcal{O}}_{P}$). The maximal ideal of ${\mathcal{O}}_{\U0001d52d}$ is $\U0001d52d{({R}_{P})}_{\U0001d52d}$.

Let ${\mathcal{O}}_{\U0001d513}$ be any prime of $L$, then it lays over some prime ideal $P$ of $K$ and in fact, if $\U0001d52d={R}_{P}\cap \U0001d513$ then ${\mathcal{O}}_{\U0001d52d}\cong {\mathcal{O}}_{\U0001d513}$. Let $\delta (\U0001d513)$ be the exact power of $\U0001d52d$ dividing the different of ${R}_{P}$ over ${\mathcal{O}}_{P}$ (the different of an extension^{} of Dedekind domains^{} is a fractional ideal^{}). We define the different divisor of $L/K$ as follows:

$${D}_{L/K}=\sum _{\U0001d513}\delta (\U0001d513)\U0001d513$$ |

as an element of the free abelian group generated by the prime ideals of $L$.

###### Theorem (Riemann-Hurwitz).

Title | Riemann-Hurwitz theorem |
---|---|

Canonical name | RiemannHurwitzTheorem |

Date of creation | 2013-03-22 15:34:40 |

Last modified on | 2013-03-22 15:34:40 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11R58 |

Defines | different divisor of an extension of function fields |