Generic Polynomial

In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if , , and are indeterminates, the generic polynomial of degree two in is $ax^2+bx+c.$

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 ''F'' is a monic polynomial ''P'' with coefficients in the field of rational functions ''L'' = ''F''(''t''₁, ..., ''t''_''n'') 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''_''n''. This is trivial, as

:$x^n + t_1 x^ + \cdots + t_n$

: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 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''_''n'' has a generic polynomial if and only if ''n'' is not divisible by eight.
* The quaternion group ''Q''₈.
* Heisenberg groups $H_$ 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.

Examples of generic polynomials

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 $gd_G$, is defined as the minimal number of parameters in a generic polynomial for ''G'' over ''F'', or $\infty$ if no generic polynomial exists.

Examples:

*$gd_A_3=1$
*$gd_S_3=1$
*$gd_D_4=2$
*$gd_S_4=2$
*$gd_D_5=2$
*$gd_{\mathbb{QS_5=2$

Publications

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

Field (mathematics)
Galois theory