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In Galois theory, a branch of modern algebra, a generic polynomial for a finite group ''G'' and field ''F'' is a monic polynomial ''P'' with coefficients in the field ''L'' = ''F''(''t''1, ..., ''t''''n'') of ''F'' with ''n'' indeterminates adjoined, such that the splitting field ''M'' of ''P'' has Galois group ''G'' over ''L'', and such that every extension ''K''/''F'' with Galois group ''G'' can be obtained as the splitting field of a polynomial which is the specialization of ''P'' resulting from setting the ''n'' indeterminates to ''n'' elements of ''F''. This is sometimes called ''F-generic'' relative to the field ''F'', with a Q-''generic'' polynomial, generic relative to the rational numbers, being called simply generic. The existence, and especially the construction, of a generic polynomial for a given Galois group provides a complete solution to the inverse Galois problem for that group. However, not all Galois groups have generic polynomials, a counterexample being the cyclic group of order eight. ==Groups with generic polynomials== * The symmetric group ''S''''n''. This is trivial, as : is a generic polynomial for ''S''''n''. * Cyclic groups ''C''''n'', where ''n'' is not divisible by eight. Lenstra showed that a cyclic group does not have a generic polynomial if ''n'' is divisible by eight, and Smith explicitly constructs such a polynomial in case ''n'' is not divisible by eight. * The cyclic group construction leads to other classes of generic polynomials; in particular the dihedral group ''D''''n'' has a generic polynomial if and only if n is not divisible by eight. * The quaternion group ''Q''8. * Heisenberg groups for any odd prime ''p''. * The alternating group ''A''4. * The alternating group ''A''5. * Reflection groups defined over Q, including in particular groups of the root systems for ''E''6, ''E''7, and ''E''8. * Any group which is a direct product of two groups both of which have generic polynomials. * Any group which is a wreath product of two groups both of which have generic polynomials. 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「generic polynomial」の詳細全文を読む スポンサード リンク
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