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

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 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

Publications

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