Noether problem

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 be the field of rational functions in variables with coefficients in the field of rational numbers, so that is a purely transcendental extension of of transcendence degree . Also, let be a finite group acting by automorphisms on by means of permutations of the variables . The question is now whether the subfield of consisting of all elements fixed under is itself a field of rational functions in (other) variables with coefficients in . 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 with a given finite group (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 was constructed in [2], and in this example is generated by a cyclic permutation of the variables. In [3] it was established that the necessary condition for the rationality of found in [2] is also sufficient. The question of rationality of in the case of an Abelian group 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 is replaced by an arbitrary field . This problem has an affirmative solution, for example, when is algebraically closed and 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 arbitrary and finite Abelian, there is a necessary and sufficient condition for rationality of (see [a1]). For example, if and is cyclic of order , then is not rational.

For , the first examples of groups for which is not rational were constructed by D.J. Saltman [a2]. He proved that for each prime number there exists such a group of order .

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