Generic polynomial

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(t₁, ..., 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

x^n + t_1 x^{n-1} + \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 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 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

Group	Generic Polynomial
C₂	$x^2-t$
C₃	$x^3-tx^2+(t-3)x+1$
S₃	$x^3-t(x+1)$
V	$(x^2-s)(x^2-t)$
C₄	$x^4-2s(t^2+1)x^2+s^2t^2(t^2+1)$
D₄	$x^4 - 2stx^2 + s^2t(t-1)$
S₄	$x^4+sx^2-t(x+1)$
D₅	$x^5+(t-3)x^4+(s-t+3)x^3+(t^2-t-2s-1)x^2+sx+t$
S₅	$x^5+sx^3-t(x+1)$

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