Generic polynomial explained

In mathematics, a generic polynomial refers usually to a polynomial whose coefficients are indeterminates. For example, if,, and are indeterminates, the generic polynomial of degree two in 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} + … + 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} ^{2t} ^{2(t} ^{2+1)}
D4	x ^{4} - 2 stx ^{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 inf ty if no generic polynomial exists.

Examples:

gd _{Q} A _{3=1} gd _{Q} S _{3=1} gd _{Q} D _{4=2} gd _{Q} S _{4=2} gd _{Q} D _{5=2} gd _{Q} S _{5=2}

Publications

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