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function field (Definition)

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

Bibliography

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



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See Also: simple transcendental field extension

Other names:  algebraic function field
Also defines:  rational function field, geometric extension, genus of a function field, degree of a prime

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Riemann-Hurwitz theorem (Theorem) by alozano
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Cross-references: infinity, ring, localization, irreducible polynomials, monic, generated by, prime ideals, dimension, degree, maximal ideal, quotient field, discrete valuation ring, prime, genus, curve, isomorphic, projective curve, nonsingular, algebraically closed, extension, finite extension, algebraic closure, fixed, algebraic extension, transcendental, polynomials, rational functions, variable, field
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This is version 5 of function field, born on 2005-11-10, modified 2008-05-19.
Object id is 7484, canonical name is FunctionField.
Accessed 6021 times total.

Classification:
AMS MSC: 11R58 (Number theory :: Algebraic number theory: global fields :: Arithmetic theory of algebraic function fields)
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