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In [[mathematics]], '''Hilbert's fourteenth problem''', that is, number 14 of [[Hilbert's problems]] proposed in 1900, asks whether certain [[ring (mathematics)|ring]]s are [[finitely generated algebra|finitely generated]].  
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The setting is as follows: Assume that ''k'' is a [[field (mathematics)|field]] and let ''K'' be a subfield of the field of [[rational function]]s in ''n'' variables,
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:''k''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> ) over ''k''.
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Consider now the ring ''R'' defined as the intersection
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:<math> R:= K \cap k[x_1, \dots, x_n] \ .</math>
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Hilbert conjectured that all such subrings are finitely generated. It can be shown that the field ''K'' is always finitely generated ''as a field'', in other words, there exist finitely many elements
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:''y''<sub>''i''</sub>, ''i'' = 1 ,...,''d'' in ''K''
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After some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for ''n'' = 1 and ''n'' = 2 by [[Zariski]] in 1954) then in 1959 [[Masayoshi Nagata]] found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a [[linear algebraic group]].
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The problem originally arose in algebraic [[invariant theory]]. Here the ring ''R'' is given as a (suitably defined) ring of polynomial invariants of a [[linear algebraic group]] over a field ''k'' acting algebraically on a [[polynomial ring]] ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] (or more generally, on a finitely generated algebra defined over a field). In this situation the field ''K'' is the field of ''rational'' functions (quotients of polynomials) in the variables ''x''<sub>''i''</sub> which are invariant under the given action of the algebraic group, the ring ''R'' is the ring of ''polynomials'' which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by [[Arthur Cayley|Cayley]], [[James Joseph Sylvester|Sylvester]], [[Alfred Clebsch|Clebsch]], [[Paul Gordan]] and also Hilbert) of invariants of [[binary quantic|binary form]]s in two variables with the natural action of the [[special linear group]] ''SL''<sub>2</sub>(''k'') on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of [[complex number]]s for some classical [[semi-simple]] [[Lie group]]s (in particular the [[general linear group]] over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming  from finite-dimensional representations of the Lie-group. This finiteness result was later extended by [[Hermann Weyl]] to the class of all semi-simple Lie-groups. A major ingredient in Hilbert's proof is the [[Hilbert basis theorem]] applied to the [[ideal (ring theory)|ideal]] inside the polynomial ring generated by the invariants.
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[[Oscar Zariski|Zariski]]'s formulation of Hilbert's fourteenth problem asks whether, for a [[Quasiprojective variety|quasi-affine]] [[algebraic variety]] ''X'' over a field ''k'', possibly assuming ''X'' [[Normal scheme|normal]] or [[Smooth (algebraic variety)|smooth]], the ring of [[regular function]]s on ''X'' is finitely generated over ''k''.
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Zariski's formulation was shown<ref>{{Citation | last = Winkelmann | first = Jörg | title = Invariant rings and quasiaffine quotients | journal = Math. Z. | volume = 244 | issue = 1 | pages = 163&ndash;174 | year = 2003 | doi = 10.1007/s00209-002-0484-9 | postscript = .}}</ref> to be equivalent to the original problem, for ''X'' normal.
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{{harvtxt|Nagata|1958}} gave the following counterexample to Hilbert's problem. The field ''k'' is a field containing 48 elements ''a''<sub>1''i''</sub>, ...,''a''<sub>16''i''</sub>, for ''i''=1, 2, 3 that are algebraically independent over the prime field. The ring ''R'' is the polynomial ring ''k''[''x''<sub>1</sub>,...,''x''<sub>16</sub>, ''t''<sub>1</sub>,...,''t''<sub>16</sub>] in 32 variables. The vector space ''V'' is a 13-dimensional vector space over ''k'' consisting of all vectors (''b''<sub>1</sub>,...,''b''<sub>16</sub>) in ''k''<sup>16</sup> orthogonal to each of the three vectors (''a''<sub>1''i''</sub>, ...,''a''<sub>16''i''</sub>) for ''i''=1, 2, 3. The vector space ''V'' is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on ''R'' by fixing all elements ''t''<sub>''j''</sub> and taking ''x''<sub>''j''</sub> to ''x''<sub>''j''</sub> + ''b''<sub>''j''</sub>''t''<sub>''j''</sub>. Then the ring of elements of ''R'' invariant under the action of the group ''V'' is not a finitely generated ''k''-algebra.
</ul>
 
Several authors have reduced the sizes of the group and the vector space in Nagata's example. For example, {{harvtxt|Totaro|2008}} showed that over any field there is an action of the sum ''G''{{su|p=3|b=a}} of three copies of the additive group on ''k''<sup>18</sup> whose ring of invariants is not finitely generated.
 
== References ==
 
*{{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Proc. Internat. Congress Math. 1958 | url=http://mathunion.org/ICM/ICM1958/ | publisher=[[Cambridge University Press]] | mr=0116056 | year=1960 | chapter=On the fourteenth problem of Hilbert | pages=459–462}}
*{{Citation | last1=Nagata | first1=Masayoshi | author1-link=Masayoshi Nagata | title=Lectures on the fourteenth problem of Hilbert | url=http://www.math.tifr.res.in/~publ/ln/tifr31.pdf | publisher=Tata Institute of Fundamental Research | location=Bombay | series=Tata Institute of Fundamental Research Lectures on Mathematics  | id={{MR|0215828}} | year=1965 | volume=31}}
*{{Citation | last1=Totaro | first1=Burt | title=Hilbert's 14th problem over finite fields and a conjecture on the cone of curves | doi=10.1112/S0010437X08003667 | mr=2457523 | year=2008 | journal=Compositio Mathematica | issn=0010-437X | volume=144 | issue=5 | pages=1176–1198}}
* O. Zariski, ''Interpretations algebrico-geometriques du quatorzieme probleme de Hilbert'', Bulletin des Sciences Mathematiques 78 (1954), pp.&nbsp;155–168.
<references/>
 
{{Hilbert's problems}}
 
[[Category:Hilbert's problems|#14]]
[[Category:Invariant theory]]

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