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"("t"₁, ..., "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 "Q"₈.

* Heisenberg groups $$H_{p^3} 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 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{QA_3=1

*$$gd_{mathbb{QS_3=1

*$$gd_{mathbb{QD_4=2

*$$gd_{mathbb{QD_5=2

Publications

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