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 [math]\displaystyle{ ax^2+bx+c. }[/math] 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

[math]\displaystyle{ x^n + t_1 x^{n-1} + \cdots + t_n }[/math]

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 [math]\displaystyle{ H_{p^3} }[/math] 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	[math]\displaystyle{ x^2-t }[/math]
C3	[math]\displaystyle{ x^3-tx^2+(t-3)x+1 }[/math]
S3	[math]\displaystyle{ x^3-t(x+1) }[/math]
V	[math]\displaystyle{ (x^2-s)(x^2-t) }[/math]
C4	[math]\displaystyle{ x^4-2s(t^2+1)x^2+s^2t^2(t^2+1) }[/math]
D4	[math]\displaystyle{ x^4 - 2stx^2 + s^2t(t-1) }[/math]
S4	[math]\displaystyle{ x^4+sx^2-t(x+1) }[/math]
D5	[math]\displaystyle{ x^5+(t-3)x^4+(s-t+3)x^3+(t^2-t-2s-1)x^2+sx+t }[/math]
S5	[math]\displaystyle{ x^5+sx^3-t(x+1) }[/math]

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 [math]\displaystyle{ gd_{F}G }[/math], is defined as the minimal number of parameters in a generic polynomial for G over F, or [math]\displaystyle{ \infty }[/math] if no generic polynomial exists.

Examples:

• [math]\displaystyle{ gd_{\mathbb{Q}}A_3=1 }[/math]
• [math]\displaystyle{ gd_{\mathbb{Q}}S_3=1 }[/math]
• [math]\displaystyle{ gd_{\mathbb{Q}}D_4=2 }[/math]
• [math]\displaystyle{ gd_{\mathbb{Q}}S_4=2 }[/math]
• [math]\displaystyle{ gd_{\mathbb{Q}}D_5=2 }[/math]
• [math]\displaystyle{ gd_{\mathbb{Q}}S_5=2 }[/math]

Publications

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