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From Wikipedia, the free encyclopedia
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
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.
Group | Generic Polynomial |
---|---|
C2 | |
C3 | |
S3 | |
V | |
C4 | |
D4 | |
S4 | |
D5 | |
S5 |
Generic polynomials are known for all transitive groups of degree 5 or less.
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:
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