Editing Generic polynomial

In [[mathematics]], a '''generic polynomial''' refers usually to a [[polynomial]] whose [[coefficient]]s are [[indeterminate (variable)|indeterminate]]s. For example, if {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are indeterminates, the generic polynomial of degree two in {{math|''x''}} is <math>ax^2+bx+c.</math>

However in [[Galois theory]], a branch of [[algebra]], and in this article, the term ''generic polynomial'' has a different, although related, meaning: a '''generic polynomial''' for a [[finite group]] ''G'' and a [[field (mathematics)|field]] ''F'' is a [[monic polynomial]] ''P'' with coefficients in the [[field of rational functions]] ''L'' = ''F''(''t''<sub>1</sub>, ..., ''t''<sub>''n''</sub>) in ''n'' indeterminates over ''F'', 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'' or relative to the field ''F''; a '''Q'''-''generic'' polynomial, which is generic relative to the rational numbers is 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''<sub>''n''</sub>. This is trivial, as

:<math>x^n + t_1 x^{n-1} + \cdots + t_n</math>

:is a generic polynomial for ''S''<sub>''n''</sub>.

* Cyclic groups ''C''<sub>''n''</sub>, where ''n'' is not [[divisibility|divisible]] by eight. [[Hendrik Lenstra|Lenstra]] showed that a cyclic group does not have a generic polynomial if ''n'' is divisible by eight, and G. W. 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''<sub>''n''</sub> has a generic polynomial if and only if ''n'' is not divisible by eight.
* The [[quaternion group]] ''Q''<sub>8</sub>.
* [[Heisenberg group]]s <math>H_{p^3}</math> for any [[parity (mathematics)|odd]] [[prime number|prime]] ''p''.
* The [[alternating group]] ''A''<sub>4</sub>.
* The alternating group ''A''<sub>5</sub>.
* Reflection groups defined over '''Q''', including in particular groups of the root systems for ''E''<sub>6</sub>, ''E''<sub>7</sub>, and ''E''<sub>8</sub>.
* Any group which is a [[direct product of groups|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.

==Examples of generic polynomials==

{| border="1" cellpadding="2"
! Group !! Generic Polynomial
|-
|''C''<sub>''2''</sub> || <math>x^2-t</math>
|-
|''C''<sub>''3''</sub> || <math>x^3-tx^2+(t-3)x+1</math>
|-
|''S''<sub>''3''</sub> || <math>x^3-t(x+1)</math>
|-
|''V'' || <math>(x^2-s)(x^2-t)</math>
|-
|''C''<sub>''4''</sub> || <math>x^4-2s(t^2+1)x^2+s^2t^2(t^2+1)</math>
|-
|''D''<sub>''4''</sub> || <math>x^4 - 2stx^2 + s^2t(t-1)</math>
|-
|''S''<sub>''4''</sub> || <math>x^4+sx^2-t(x+1)</math>
|-
|''D''<sub>''5''</sub> || <math>x^5+(t-3)x^4+(s-t+3)x^3+(t^2-t-2s-1)x^2+sx+t</math>
|-
|''S''<sub>''5''</sub> || <math>x^5+sx^3-t(x+1)</math>
|}

Generic polynomials are known for all transitive groups of degree 5 or less.

==Generic dimension==

The '''generic dimension''' for a finite group ''G'' over a field ''F'', denoted <math>gd_{F}G</math>, is defined as the minimal number of parameters in a generic polynomial for ''G'' over ''F'', or <math>\infty</math> if no generic polynomial exists.

Examples:

*<math>gd_{\mathbb{Q}}A_3=1</math>
*<math>gd_{\mathbb{Q}}S_3=1</math>
*<math>gd_{\mathbb{Q}}D_4=2</math>
*<math>gd_{\mathbb{Q}}S_4=2</math>
*<math>gd_{\mathbb{Q}}D_5=2</math>
*<math>gd_{\mathbb{Q}}S_5=2</math>

==Publications==

*Jensen, Christian U., Ledet, Arne, and Yui, Noriko, ''Generic Polynomials'', Cambridge University Press, 2002

[[Category:Field (mathematics)]]
[[Category:Galois theory]]