function field

Let $F$ be a field.

Definition 1.

The rational function field over $F$ in one variable ( $x$ ), denoted by $F(x)$ , is the field of all rational functions $p(x)/q(x)$ with polynomials $p,q\in F[x]$ and $q(x)$ 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 $K/F(x)$ 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 $L/K$ of function fields is said to be geometric if $L\cap\overline{F}=F$ .

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

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 $F\subset R$ 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 $\deg P$ , is defined to be the dimension of $R/P$ over $F$ .

Example 2.

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

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

References

1 R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York.
2 M. Rosen, Number Theory in Function Fields, Springer-Verlag, New York.

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