The question of the rationality of the field of invariants of a finite group acting by automorphisms on a field of rational functions. More precisely, let $K=\mathbf Q(x_1,\dots,x_n)$ be the field of rational functions in $n$ variables with coefficients in the field $\mathbf Q$ of rational numbers, so that $K$ is a purely transcendental extension of $\mathbf Q$ of transcendence degree $n$. Also, let $G$ be a finite group acting by automorphisms on $K$ by means of permutations of the variables $x_1,\dots,x_n$. The question is now whether the subfield $K^G$ of $K$ consisting of all elements fixed under $G$ is itself a field of rational functions in $n$ (other) variables with coefficients in $\mathbf Q$. This question was raised by E. Noether  in connection with the inverse problem of Galois theory (cf. Galois theory, inverse problem of). If the answer to Noether's problem were affirmative, one could construct a Galois extension of $\mathbf Q$ with a given finite group $G$ (see ). The problem is also closely connected with the Lüroth problem.
In general, the answer to Noether's problem is negative. The first example of a non-rational field $K^G$ was constructed in , and in this example $G$ is generated by a cyclic permutation of the variables. In  it was established that the necessary condition for the rationality of $K^G$ found in  is also sufficient. The question of rationality of $K^G$ in the case of an Abelian group $G$ is closely connected with the theory of algebraic tori (cf. Algebraic torus) (see ).
Frequently, Noether's problem is interpreted more generally as the problem that arises when in the original setting $\mathbf Q$ is replaced by an arbitrary field $k$. This problem has an affirmative solution, for example, when $k$ is algebraically closed and $G$ is Abelian.
|||E. Noether, "Gleichungen mit vorgeschriebener Gruppe" Math. Ann. , 78 (1917–1918) pp. 221–229|
|||R.G. Swan, "Invariant rational functions and a problem of Steenrod" Invent. Math. , 7 : 2 (1969) pp. 148–158|
|||V.E. Voskresenskii, "Rationality of certain algebraic tori" Math. USSR. Izv. , 35 : 5 (1979) pp. 1049–1056 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 1037–1046|
|||V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian)|
|||N.G. Chebotarev, "Grundzüge der Galois'schen Theorie" , Noordhoff (1950) pp. Chapt. V §4 (Translated from Russian)|
For $k$ arbitrary and $G$ finite Abelian, there is a necessary and sufficient condition for rationality of $K^G$ (see [a1]). For example, if $k=\mathbf Q$ and $G$ is cyclic of order $8$, then $K^G$ is not rational.
For $k=\mathbf C$, the first examples of groups $G$ for which $K^G$ is not rational were constructed by D.J. Saltman [a2]. He proved that for each prime number $p$ there exists such a group of order $p^9$.
|[a1]||H.W. Lenstra Jr., "Rational functions invariant under a finite abelian group" Invent. Math. , 25 (1974) pp. 299–325|
|[a2]||D.J. Saltman, "Noether's problem over an algebraically closed field" Invent. Math. , 77 (1984) pp. 71–84|
Noether problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noether_problem&oldid=34909