function field

Let $F$ be a field.

Definition 1.

The rational function field over $F$ in one variable ($x$), denoted by $F\mathit{}\mathrm{(}x\mathrm{)}$, is the field of all rational functions $p\mathit{}\mathrm{(}x\mathrm{)}\mathrm{/}q\mathit{}\mathrm{(}x\mathrm{)}$ with polynomials $p\mathrm{,}q\mathrm{\in }F\mathit{}\mathrm{[}x\mathrm{]}$ and $q\mathit{}\mathrm{(}x\mathrm{)}$ not identically zero.

Definition 2.

A function field (in one variable) over $F$ is a field $K$, containing $F$ and at least one element $x$, transcendental over $F$, such that $K\mathrm{/}F\mathit{}\mathrm{(}x\mathrm{)}$ is a http://planetmath.org/node/FiniteExtensionfinite algebraic extension.

Let $\overline{F}$ be a fixed algebraic closure of $F$.

Definition 3.

Let $K$ be a function field over $F$ and let $L$ be a finite extension of $K$. The extension $L\mathrm{/}K$ of function fields is said to be geometric if $L\mathrm{\cap }\overline{F}\mathrm{=}F$.

Example 1.

The extension $\mathbb{Q}(\sqrt{x})/\mathbb{Q}(x)$ is geometric, but $\mathbb{Q}(\sqrt{2})(x)/\mathbb{Q}(x)$ is not geometric.

Theorem 1 (Thm. I.6.9 of [1]).

Let $K$ be a function field over an algebraically closed field $F$. There exists a nonsingular projective curve $C_K$ such that the function field of $C_K$ is isomorphic to $K$.

Definition 4.

Let $K$ be a function field over a field $F$. Let $K^{\mathrm{\prime }}\mathrm{=}K\mathit{}\overline{F}$ which is a function field over $\overline{F}$, a fixed algebraic closure of $F$, and let $C_{K^{\mathrm{\prime }}}$ be the curve given by the previous theorem. The genus of $K$ is, by definition, the genus of $C_{K^{\mathrm{\prime }}}$.

Definition 5.

Let $K$ be a function field over a field $F$. A prime in $K$ is by definition a discrete valuation ring $R$ with maximal $P$ such that $F\mathrm{\subset }R$ and the quotient field of $R$ is equal to $K$. The prime is usually denoted $P$ after the maximal ideal of $R$. The degree of $P$, denoted by $\mathrm{deg}\mathit{}P$, is defined to be the dimension of $R\mathrm{/}P$ over $F$.

Example 2.

Let $K=F(x)$ be the rational function field over $F$ and let $𝒪=F[x]$. The prime ideals of $𝒪$ are generated by monic irreducible polynomials in $F[x]$. Let $P=(f(x))$ be such a prime. Then $R_P={𝒪}_P$, the localization of $𝒪$ at the prime $P$ is a discrete valuation ring with $F\subset {𝒪}_P$ and the quotient field of $R_P$ is $K$. Thus $R_P={𝒪}_P$ is a prime of $K$.

One can define an ‘extra’ prime in the following way. Let $R_{\mathrm{\infty }}={𝒪}_{\mathrm{\infty }}=F[\frac{1}{x}]$ and let $P_{\mathrm{\infty }}=(\frac{1}{x})$ be the prime ideal of $R_{\mathrm{\infty }}$ generated by $\frac{1}{x}$. The localization ring ${(R_{\mathrm{\infty }})}_{P_{\mathrm{\infty }}}$ is a prime of $K$, called the prime at infinity.

Title	function field
Canonical name	FunctionField
Date of creation	2013-03-22 15:34:35
Last modified on	2013-03-22 15:34:35
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	8
Author	alozano (2414)
Entry type	Definition
Classification	msc 11R58
Synonym	algebraic function field
Related topic	SimpleTranscendentalFieldExtension
Defines	rational function field
Defines	geometric extension
Defines	genus of a function field
Defines	degree of a prime