Generic polynomial

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 formula_1
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"("t"1, ..., "t""n") 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.
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 formula_4, is defined as the minimal number of parameters in a generic polynomial for "G" over "F", or formula_5 if no generic polynomial exists.
Examples: