Chemistry - Generic polynomial

In Galois theory, 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	$x 3 - t x - (t - 3) x - 1$
C₄	$x 4 - 2 s (t 2 + 1) + s 2 t 2 (t 2 + 1)$
D₄	$x 4 - 2 s t x 2 + s 2 t (t - 1)$

Publications

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

Categories: Field theory | Galois theory