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Encyclopedia > Function field

In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions.

The ring of regular functions on a variety V defined over a field K is an integral domain if and only if the variety is irreducible, and in this case the field of fractions is defined. It is a field extension of the ground field K; its transcendence degree is by definition the dimension of the variety. All extensions of K that are finitely-generated as fields arise in this way from some algebraic variety.

In the particular case of an algebraic curve C, that is, dimension 1, it follows that any two non-constant functions F and G on C satisfy a polynomial equation P(F,G) = 0.

Properties of the variety V that depend only on the function field are studied in birational geometry. Results from FactBites:

 Function Fields (1849 words) The function field of the curve is the corresponding field of fractions. This is a vector subspace of the function field of a curve. Since function fields are realised by completely separate code, one cannot automatically expect rational functions written in terms of the generators of the coordinate ring of the curve to be elements of the function field.
 Function Fields (2592 words) The function field of the curve is the corresponding field of fractions in the affine case and the homogeneous degree 0 part of this in projective cases. As with schemes generally, a function field is attached to projective curves and the same object represents the function field of all of its affine patchs. Return the function f in the function field of a scheme as a function in projective coordinates (as an element in the field of fractions of the coordinate ring of the projective scheme having function field the parent of f).
More results at FactBites »

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