Noether problem

2020 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]

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 [1] 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 [5]). 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 [2], and in this example $G$ is generated by a cyclic permutation of the variables. In [3] it was established that the necessary condition for the rationality of $K^G$ found in [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) (see [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

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

Comments

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 $p^9$.

References

[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