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 a x 2 + b x + c . {\displaystyle 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, ..., tn) 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 Sn. This is trivial, as
x n + t 1 x n − 1 + ⋯ + t n {\displaystyle x^{n}+t_{1}x^{n-1}+\cdots +t_{n}}
is a generic polynomial for Sn.
  • Cyclic groups Cn, 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 Dn has a generic polynomial if and only if n is not divisible by eight.
  • The quaternion group Q8.
  • Heisenberg groups H p 3 {\displaystyle 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 {\displaystyle x^{2}-t}
C3 x 3 − t x 2 + ( t − 3 ) x + 1 {\displaystyle x^{3}-tx^{2}+(t-3)x+1}
S3 x 3 − t ( x + 1 ) {\displaystyle x^{3}-t(x+1)}
V ( x 2 − s ) ( x 2 − t ) {\displaystyle (x^{2}-s)(x^{2}-t)}
C4 x 4 − 2 s ( t 2 + 1 ) x 2 + s 2 t 2 ( t 2 + 1 ) {\displaystyle x^{4}-2s(t^{2}+1)x^{2}+s^{2}t^{2}(t^{2}+1)}
D4 x 4 − 2 s t x 2 + s 2 t ( t − 1 ) {\displaystyle x^{4}-2stx^{2}+s^{2}t(t-1)}
S4 x 4 + s x 2 − t ( x + 1 ) {\displaystyle x^{4}+sx^{2}-t(x+1)}
D5 x 5 + ( t − 3 ) x 4 + ( s − t + 3 ) x 3 + ( t 2 − t − 2 s − 1 ) x 2 + s x + t {\displaystyle x^{5}+(t-3)x^{4}+(s-t+3)x^{3}+(t^{2}-t-2s-1)x^{2}+sx+t}
S5 x 5 + s x 3 − t ( x + 1 ) {\displaystyle 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 g d F G {\displaystyle gd_{F}G} , is defined as the minimal number of parameters in a generic polynomial for G over F, or ∞ {\displaystyle \infty } if no generic polynomial exists.

Examples:

  • g d Q A 3 = 1 {\displaystyle gd_{\mathbb {Q} }A_{3}=1}
  • g d Q S 3 = 1 {\displaystyle gd_{\mathbb {Q} }S_{3}=1}
  • g d Q D 4 = 2 {\displaystyle gd_{\mathbb {Q} }D_{4}=2}
  • g d Q S 4 = 2 {\displaystyle gd_{\mathbb {Q} }S_{4}=2}
  • g d Q D 5 = 2 {\displaystyle gd_{\mathbb {Q} }D_{5}=2}
  • g d Q S 5 = 2 {\displaystyle gd_{\mathbb {Q} }S_{5}=2}

Publications

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