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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667801.png" /> be the field of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667802.png" /> variables with coefficients in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667803.png" /> of rational numbers, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667804.png" /> is a purely [[Transcendental extension|transcendental extension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667805.png" /> of transcendence degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667806.png" />. Also, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667807.png" /> be a [[Finite group|finite group]] acting by automorphisms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667808.png" /> by means of permutations of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n0667809.png" />. The question is now whether the subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678011.png" /> consisting of all elements fixed under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678012.png" /> is itself a field of rational functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678013.png" /> (other) variables with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678015.png" /> with a given finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678016.png" /> (see [[#References|[5]]]). The problem is also closely connected with the [[Lüroth problem|Lüroth 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678017.png" /> was constructed in [[#References|[2]]], and in this example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678018.png" /> is generated by a cyclic permutation of the variables. In [[#References|[3]]] it was established that the necessary condition for the rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678019.png" /> found in [[#References|[2]]] is also sufficient. The question of rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678020.png" /> in the case of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678021.png" /> is closely connected with the theory of algebraic tori (cf. [[Algebraic torus|Algebraic torus]]) (see [[#References|[4]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678022.png" /> is replaced by an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678023.png" />. This problem has an affirmative solution, for example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678024.png" /> is algebraically closed and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678025.png" /> is Abelian.
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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.
  
 
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====Comments====
 
====Comments====
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678026.png" /> arbitrary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678027.png" /> finite Abelian, there is a necessary and sufficient condition for rationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678028.png" /> (see [[#References|[a1]]]). For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678030.png" /> is cyclic of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678032.png" /> is not rational.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678033.png" />, the first examples of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678034.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678035.png" /> is not rational were constructed by D.J. Saltman [[#References|[a2]]]. He proved that for each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678036.png" /> there exists such a group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066780/n06678037.png" />.
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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====
 
====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>
 
<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>

Revision as of 05:08, 16 September 2014

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
How to Cite This Entry:
Noether problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noether_problem&oldid=11427
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article