In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if a , b , and c are indeterminates, the generic polynomial of degree two in x is ax^2+bx+c. 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 group G and a field F is a monic polynomial P with coefficients in the field of rational functions L = F(t1, ..., tn) in n indeterminates over F, 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 or relative to the field F; 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.

Groups with 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 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:

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