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- Hilbert's_fourteenth_problem abstract "In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated. The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables, k(x1, ..., xn ) over k.Consider now the ring R defined as the intersection 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 yi, i = 1 ,...,d in Ksuch that every element in R can be rationally represented by the yi. But this does not imply that the ring R is finitely generated as a ring, even if all the elements yi could be chosen from R. 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.".
- Hilbert's_fourteenth_problem wikiPageExternalLink ICM1958.
- Hilbert's_fourteenth_problem wikiPageExternalLink tifr31.pdf.
- Hilbert's_fourteenth_problem wikiPageID "1695231".
- Hilbert's_fourteenth_problem wikiPageRevisionID "543983617".
- Hilbert's_fourteenth_problem b "a".
- Hilbert's_fourteenth_problem hasPhotoCollection Hilbert's_fourteenth_problem.
- Hilbert's_fourteenth_problem p "3".
- Hilbert's_fourteenth_problem subject Category:Hilbert's_problems.
- Hilbert's_fourteenth_problem subject Category:Invariant_theory.
- Hilbert's_fourteenth_problem type Abstraction100002137.
- Hilbert's_fourteenth_problem type Attribute100024264.
- Hilbert's_fourteenth_problem type Condition113920835.
- Hilbert's_fourteenth_problem type Difficulty114408086.
- Hilbert's_fourteenth_problem type Hilbert'sProblems.
- Hilbert's_fourteenth_problem type Problem114410605.
- Hilbert's_fourteenth_problem type State100024720.
- Hilbert's_fourteenth_problem comment "In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain rings are finitely generated. The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables, k(x1, ..., xn ) over k.Consider now the ring R defined as the intersection Hilbert conjectured that all such subrings are finitely generated.".
- Hilbert's_fourteenth_problem label "Hilbert's fourteenth problem".
- Hilbert's_fourteenth_problem label "Четырнадцатая проблема Гильберта".
- Hilbert's_fourteenth_problem label "希爾伯特第十四問題".
- Hilbert's_fourteenth_problem sameAs m.05p17t.
- Hilbert's_fourteenth_problem sameAs Q4515163.
- Hilbert's_fourteenth_problem sameAs Q4515163.
- Hilbert's_fourteenth_problem sameAs Hilbert's_fourteenth_problem.
- Hilbert's_fourteenth_problem wasDerivedFrom Hilbert's_fourteenth_problem?oldid=543983617.
- Hilbert's_fourteenth_problem isPrimaryTopicOf Hilbert's_fourteenth_problem.