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Concept
Rational variety
Summary
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to
the field of all rational functions for some set of indeterminates, where d is the dimension of the variety.
Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in . If V is rational, then there are n + 1 polynomials g0, ..., gn in such that In order words, we have a of the variety.
Conversely, such a rational parameterization induces a field homomorphism of the field of functions of V into . But this homomorphism is not necessarily onto. If such a parameterization exists, the variety is said to be unirational. Lüroth's theorem (see below) implies that unirational curves are rational. Castelnuovo's theorem implies also that, in characteristic zero, every unirational surface is rational.
A rationality question asks whether a given field extension is rational, in the sense of being (up to isomorphism) the function field of a rational variety; such field extensions are also described as purely transcendental. More precisely, the rationality question for the field extension is this: is isomorphic to a rational function field over in the number of indeterminates given by the transcendence degree?
There are several different variations of this question, arising from the way in which the fields and are constructed.
For example, let be a field, and let
be indeterminates over K and let L be the field generated over K by them. Consider a finite group permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of , typically denoted . The rationality question for is called Noether's problem and asks if this field of fixed points is or is not a purely transcendental extension of K.
In the paper on Galois theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to "Noether's problem".
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