# Generic polynomial

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(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.

## Groups with generic polynomials

* The symmetric group S$_n$. This is trivial, as
    $x^{n}+t_{1}x^{n-1}+\cdots +t_{n}$
    is a generic polynomial for S$_n$.
* Cyclic groups C$_n$, where n is not divisible by eight. Lenstra showed that a cyclic group does not have a generic polynomial if n is divisible by eight, and G. W. Smith explicitly constructs such a polynomial in case n is not divisible by eight.
* The cyclic group construction leads to other classes of generic polynomials; in particular the dihedral group D$_n$ has a generic polynomial if and only if n is not divisible by eight.
* The quaternion group Q$_8$.
* Heisenberg groups $H_{p^{3}}$ for any odd prime p.
* The alternating group A$_4$.
* The alternating group A$_5$.
* Reflection groups defined over Q, including in particular groups of the root systems for E$_6$, E$_7$, and E$_8$.
* Any group which is a direct product of two groups both of which have generic polynomials.
* Any group which is a wreath product of two groups both of which have generic polynomials.

## Examples of generic polynomials

| Group | Generic Polynomial |
| C$_2$ | $x^{2}-t$ |
| C$_3$ | $x^{3}-tx^{2}+(t-3)x+1$ |
| S$_3$ | $x^{3}-t(x+1)$ |
| V | $(x^{2}-s)(x^{2}-t)$ |
| C$_4$ | $x^{4}-2s(t^{2}+1)x^{2}+s^{2}t^{2}(t^{2}+1)$ |
| D$_4$ | $x^{4}-2stx^{2}+s^{2}t(t-1)$ |
| S$_4$ | $x^{4}+sx^{2}-t(x+1)$ |
| D$_5$ | $x^{5}+(t-3)x^{4}+(s-t+3)x^{3}+(t^{2}-t-2s-1)x^{2}+sx+t$ |
| S$_5$ | $x^{5}+sx^{3}-t(x+1)$ |
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:
* $gd_{\mathbb {Q} }A_{3}=1$
* $gd_{\mathbb {Q} }S_{3}=1$
* $gd_{\mathbb {Q} }D_{4}=2$
* $gd_{\mathbb {Q} }S_{4}=2$
* $gd_{\mathbb {Q} }D_{5}=2$
* $gd_{\mathbb {Q} }S_{5}=2$

## Publications

* Jensen, Christian U., Ledet, Arne, and Yui, Noriko, Generic Polynomials, Cambridge University Press, 2002