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Rationality question

In mathematics, a rationality question asks whether a given field extension is rational in the sense of algebraic geometry; such field extensions are also described as purely transcendental. More precisely, the rationality question for the field extension $K < L$ is this: is $L$ isomorphic to the rational function field

K (x 1,..., x n),

for some set ${x 1,..., x n}$ of indeterminates?

There are several different variations of this question, arising from the way in which the fields $K$ and $L$ are constructed.

For example, let $K$ be a field, and let

{y 1,... y m}

be indeterminates. Consider a finite group $G$ permuting those indeterminates. By standard Galois theory, the set of fixed points of this group action is a subfield of $L$ , typically denoted $L G$ . The rationality question for $K < L G$ is called Noether's problem .

10-26-2009 08:16:03
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