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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated
field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
''K''/''k'' which has
transcendence degree In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients ...
''n'' over ''k''. Equivalently, an algebraic function field of ''n'' variables over ''k'' may be defined as a finite field extension of the field ''K'' = ''k''(''x''1,...,''x''''n'') of rational functions in ''n'' variables over ''k''.


Example

As an example, in the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
''k'' 'X'',''Y''consider the ideal generated by the
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
''Y''2 − ''X''3 and form the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the fie ...
of the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
''k'' 'X'',''Y''(''Y''2 − ''X''3). This is a function field of one variable over ''k''; it can also be written as k(X)(\sqrt) (with degree 2 over k(X)) or as k(Y)(\sqrt (with degree 3 over k(Y)). We see that the degree of an algebraic function field is not a well-defined notion.


Category structure

The algebraic function fields over ''k'' form a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
; the
morphisms In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
from function field ''K'' to ''L'' are the
ring homomorphism In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
s ''f'' : ''K'' → ''L'' with ''f''(''a'') = ''a'' for all ''a'' in ''k''. All these morphisms are
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
. If ''K'' is a function field over ''k'' of ''n'' variables, and ''L'' is a function field in ''m'' variables, and ''n'' > ''m'', then there are no morphisms from ''K'' to ''L''.


Function fields arising from varieties, curves and Riemann surfaces

The
function field of an algebraic variety In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects that are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorp ...
of dimension ''n'' over ''k'' is an algebraic function field of ''n'' variables over ''k''. Two varieties are birationally equivalent if and only if their function fields are isomorphic. (But note that non-
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
varieties may have the same function field!) Assigning to each variety its function field yields a duality (contravariant equivalence) between the category of varieties over ''k'' (with dominant rational maps as morphisms) and the category of algebraic function fields over ''k''. (The varieties considered here are to be taken in the scheme sense; they need not have any ''k''-rational points, like the curve defined over the reals, that is with .) The case ''n'' = 1 (irreducible algebraic curves in the scheme sense) is especially important, since every function field of one variable over ''k'' arises as the function field of a uniquely defined regular (i.e. non-singular) projective irreducible algebraic curve over ''k''. In fact, the function field yields a duality between the category of regular projective irreducible algebraic curves (with dominant regular maps as morphisms) and the category of function fields of one variable over ''k''. The field M(''X'') of
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the function. ...
s defined on a connected
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
''X'' is a function field of one variable over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s C. In fact, M yields a duality (contravariant equivalence) between the category of compact connected Riemann surfaces (with non-constant holomorphic maps as morphisms) and function fields of one variable over C. A similar correspondence exists between compact connected
Klein surface In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves ov ...
s and function fields in one variable over R.


Number fields and finite fields

The function field analogy states that almost all theorems on
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s have a counterpart on function fields of one variable over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
, and these counterparts are frequently easier to prove. (For example, see Analogue for irreducible polynomials over a finite field.) In the context of this analogy, both number fields and function fields over finite fields are usually called "
global field In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...
s". The study of function fields over a finite field has applications in
cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
and
error correcting code In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The centra ...
s. For example, the function field of an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...
over a finite field (an important mathematical tool for
public key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic al ...
) is an algebraic function field. Function fields over the field of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s play also an important role in solving
inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...
s.


Field of constants

Given any algebraic function field ''K'' over ''k'', we can consider the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of elements of ''K'' which are algebraic over ''k''. These elements form a field, known as the ''field of constants'' of the algebraic function field. For instance, C(''x'') is a function field of one variable over R; its field of constants is C.


Valuations and places

Key tools to study algebraic function fields are absolute values, valuations, places and their completions. Given an algebraic function field ''K''/''k'' of one variable, we define the notion of a ''valuation ring'' of ''K''/''k'': this is a
subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
''O'' of ''K'' that contains ''k'' and is different from ''k'' and ''K'', and such that for any ''x'' in ''K'' we have ''x'' ∈ ''O'' or ''x'' -1 ∈ ''O''. Each such valuation ring is a discrete valuation ring and its maximal ideal is called a ''place'' of ''K''/''k''. A ''discrete valuation'' of ''K''/''k'' is a
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
function ''v'' : ''K'' → Z∪ such that ''v''(x) = ∞ iff ''x'' = 0, ''v''(''xy'') = ''v''(''x'') + ''v''(''y'') and ''v''(''x'' + ''y'') ≥ min(''v''(''x''),''v''(''y'')) for all ''x'', ''y'' ∈ ''K'', and ''v''(''a'') = 0 for all ''a'' ∈ ''k'' \ . There are natural bijective correspondences between the set of valuation rings of ''K''/''k'', the set of places of ''K''/''k'', and the set of discrete valuations of ''K''/''k''. These sets can be given a natural
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
structure: the
Zariski–Riemann space In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a c ...
of ''K''/''k''.


See also

*
function field of an algebraic variety In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects that are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorp ...
*
function field (scheme theory) The sheaf of rational functions ''KX'' of a scheme ''X'' is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties, such a sheaf associates to ...
* algebraic function * Drinfeld module


References

{{reflist Field (mathematics)