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.

单词	FunctionField
释义	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.
随便看	monodromy theorem Monoid monoid MonoidalCategory monoidal category MonoidBialgebra monoid bialgebra MonoidBialgebraIsAHopfAlgebraIfAndOnlyIfMonoidIsAGroup monoid bialgebra is a Hopf algebra if and only if monoid is a group Monomial monomial MonomialMatrix monomial matrix MonotoneClass monotone class MonotoneClassTheorem monotone class theorem MonotoneConvergenceTheorem monotone convergence theorem Monotonic monotonic MonotonicallyDecreasing monotonically decreasing MonotonicallyIncreasing monotonically increasing