Rational Variety - Rationality Questions

Rationality Questions

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 (Noether 1918) on Galois theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to "Noether's problem". (She first mentioned this problem in (Noether 1913) where she attributed the problem to E. Fischer.) She showed this was true for n = 2, 3, or 4. R. G. Swan (1969) found a counter-example to the Noether's problem, with n = 47 and G a cyclic group of order 47.

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