Generic polynomial

In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if $a$, $b$, and $c$ are indeterminates, the generic polynomial of degree two in $x$ 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(t1, ..., tn) 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 Sn. This is trivial, as


 * $$x^n + t_1 x^{n-1} + \cdots + t_n$$


 * is a generic polynomial for Sn.


 * Cyclic groups Cn, 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 Dn has a generic polynomial if and only if n is not divisible by eight.
 * The quaternion group Q8.
 * Heisenberg groups $$H_{p^3}$$ for any odd prime p.
 * The alternating group A4.
 * The alternating group A5.
 * Reflection groups defined over Q, including in particular groups of the root systems for E6, E7, and E8.
 * 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 $$gd_{F}G$$, is defined as the minimal number of parameters in a generic polynomial for G over F, or $$\infty$$ if no generic polynomial exists.

Examples:


 * $$gd_{\mathbb{Q}}A_3=1$$
 * $$gd_{\mathbb{Q}}S_3=1$$
 * $$gd_{\mathbb{Q}}D_4=2$$
 * $$gd_{\mathbb{Q}}S_4=2$$
 * $$gd_{\mathbb{Q}}D_5=2$$
 * $$gd_{\mathbb{Q}}S_5=2$$

Publications

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