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.

Read more about Generic Polynomial: Groups With Generic Polynomials, Examples of Generic Polynomials, Generic Dimension, Publications

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