View source for Noether problem

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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|transcendental extension]] of $\mathbf Q$ of transcendence degree $n$. Also, let $G$ be a [[Finite group|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 [[#References|[1]]] in connection with the inverse problem of Galois theory (cf. [[Galois theory, inverse problem of|Galois theory, inverse problem of]]). If the answer to Noether's problem were affirmative, one could construct a [[Galois extension|Galois extension]] of $\mathbf Q$ with a given finite group $G$ (see [[#References|[5]]]). The problem is also closely connected with the [[Lüroth problem|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 [[#References|[2]]], and in this example $G$ is generated by a cyclic permutation of the variables. In [[#References|[3]]] it was established that the necessary condition for the rationality of $K^G$ found in [[#References|[2]]] 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|Algebraic torus]]) (see [[#References|[4]]]).

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Noether,   "Gleichungen mit vorgeschriebener Gruppe"  ''Math. Ann.'' , '''78'''  (1917–1918)  pp. 221–229</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.G. Swan,   "Invariant rational functions and a problem of Steenrod"  ''Invent. Math.'' , '''7''' :  2  (1969)  pp. 148–158</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.E. Voskresenskii,   "Algebraic tori" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chebotarev,   "Grundzüge der Galois'schen Theorie" , Noordhoff  (1950)  pp. Chapt. V §4  (Translated from Russian)</TD></TR></table>



====Comments====
For $k$ arbitrary and $G$ finite Abelian, there is a necessary and sufficient condition for rationality of $K^G$ (see [[#References|[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 [[#References|[a2]]]. He proved that for each prime number $p$ there exists such a group of order $p^9$.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.W. Lenstra Jr.,   "Rational functions invariant under a finite abelian group"  ''Invent. Math.'' , '''25'''  (1974)  pp. 299–325</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.J. Saltman,   "Noether's problem over an algebraically closed field"  ''Invent. Math.'' , '''77'''  (1984)  pp. 71–84</TD></TR></table>