|
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory. == Formulation of the theorem == Hilbert's irreducibility theorem. Let : be irreducible polynomials in the ring : Then there exists an ''r''-tuple of rational numbers (''a''1,...,''a''''r'') such that : are irreducible in the ring : Remarks. * It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible specialization, called Hilbert set, is large in many senses. For example, this set is Zariski dense in . * There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (''a''1,...,''a''''r'') to be integers. * There are many Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example, global fields are Hilbertian.〔Lang (1997) p.41〕 * The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take in the definition. A recent result of Bary-Soroker shows that for a field ''K'' to be Hilbertian it suffices to consider the case of and absolutely irreducible, that is, irreducible in the ring ''K''alg(), where ''K''alg is the algebraic closure of ''K''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert's irreducibility theorem」の詳細全文を読む スポンサード リンク
|