Generic polynomial

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(t1, ..., tn) 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.

Groups with generic polynomials

x^n + t_1 x^{n-1} + \cdots + t_n

is a generic polynomial for Sn.

Examples of generic polynomials

Group Generic Polynomial
C2 x^2-t
C3 x^3-tx^2+(t-3)x+1
S3 x^3-t(x+1)
V (x^2-s)(x^2-t)
C4 x^4-2s(t^2+1)x^2+s^2t^2(t^2+1)
D4 x^4 - 2stx^2 + s^2t(t-1)
S4 x^4+sx^2-t(x+1)
D5 x^5+(t-3)x^4+(s-t+3)x^3+(t^2-t-2s-1)x^2+sx+t
S5 x^5+sx^3-t(x+1)

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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