In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if,, and are indeterminates, the generic polynomial of degree two in is 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 groupG and a fieldF is a monic polynomialP with coefficients in the field of rational functionsL = F in n indeterminates over F, such that the splitting fieldM of P has Galois groupG 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 fieldF; 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.
Cyclic groupsCn, 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 groupDn has a generic polynomial if and only if n is not divisible by eight.
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, is defined as the minimal number of parameters in a generic polynomial for G over F, or if no generic polynomial exists. Examples:
Publications
Jensen, Christian U., Ledet, Arne, and Yui, Noriko, Generic Polynomials, Cambridge University Press, 2002
