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Zariski's Finiteness Theorem
In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case.http://aix1.uottawa.ca/~ddaigle/articles/H14survey.pdf Precisely, it states: :Given a normal domain ''A'', finitely generated as an algebra over a field ''k'', if ''L'' is a subfield of the field of fractions of ''A'' containing ''k'' such that \operatorname_k(L) \le 2, then the ''k''-subalgebra L \cap A is finitely generated. References * Hilbert's problems Invariant theory Commutative algebra {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's 14th Problem
In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras 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''(''x''1, ..., ''x''''n'' ) over ''k''. Consider now the ''k''-algebra ''R'' defined as the intersection : R:= K \cap k _1, \dots, x_n\ . Hilbert conjectured that all such algebras are finitely generated over ''k''. 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. History The problem originally arose in algebraic invari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Domain
In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root of a monic polynomial with coefficients in ''A,'' then ''x'' is itself an element of ''A.'' Many well-studied domains are integrally closed: fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. Note that integrally closed domains appear in the following chain of class inclusions: Basic properties Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a field extension of ''K''. Then ''x''∈''L'' is integral over ''A'' if and only if it is algebraic over ''K'' and its minimal polynomial over ''K'' has coefficients in ''A''. In particular, this means that any element of ''L'' integral over ''A'' is root of a monic polynomial in ''A'' 'X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the University of Paris, Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in and Nature and influence of the problems Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. Introduction Let G be a group, and V a finite-dimensional vector space over a field k (which in classical invariant theory was usually assumed to be the complex numbers). A representation of G in V is a group homomorphism \pi:G \to GL(V), which induces a group action of G on V. If k /math> is the space of polynomial functions on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
