Generic polynomial

Last updated February 15, 2024

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(t₁, ..., 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 Q₈.
Heisenberg groups $H_{p^{3}}$ for any odd prime p.
The alternating group A₄.
The alternating group A₅.
Reflection groups defined over Q, including in particular groups of the root systems for E₆, E₇, and E₈.
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
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	$x^{4}-2s(t^{2}+1)x^{2}+s^{2}t^{2}(t^{2}+1)$
D₄	$x^{4}-2stx^{2}+s^{2}t(t-1)$
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:

$gd_{\mathbb {Q} }A_{3}=1$
$gd_{\mathbb {Q} }S_{3}=1$
$gd_{\mathbb {Q} }D_{4}=2$
$gd_{\mathbb {Q} }S_{4}=2$
$gd_{\mathbb {Q} }D_{5}=2$
$gd_{\mathbb {Q} }S_{5}=2$

Publications

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

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