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(x1,...,xn),
for some set {x1,...,xn} 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
- {y1,...ym}
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 LG. The rationality question for K < LG is called Noether's problem .
