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Concept
Algebraic function field
Summary
In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree 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(x1,...,xn) of rational functions in n variables over k.
As an example, in the polynomial ring k [X,Y] consider the ideal generated by the irreducible polynomial Y 2 − X 3 and form the field of fractions of the quotient ring k [X,Y]/(Y 2 − X 3). This is a function field of one variable over k; it can also be written as (with degree 2 over ) or as (with degree 3 over ). We see that the degree of an algebraic function field is not a well-defined notion.
The algebraic function fields over k form a ; the from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. 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.
The function field of an algebraic variety 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 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 X2 + Y2 + 1 = 0 defined over the reals, that is with k = R.)
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.
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