Generic polynomial

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In Galois theory, a generic polynomial for a finite group G and field F is a monic polynomial P with coefficients in the field L = F(t1, ..., tn) of F with n indeterminates adjoined, 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 relative to the field F, with a Q-generic polynomial, generic relative to the rational numbers, being 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.

[edit] Groups with generic polynomials

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 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 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.

[edit] Examples of generic polynomials

Group Generic Polynomial
C2 x2 − t
C3 x3 − tx − (t − 3)x − 1
C4 x4 − 2s(t2 + 1) + s2t2(t2 + 1)
D4 x4 − 2stx2 + s2t(t − 1)

[edit] Publications

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