Generic polynomial

• The symmetric group Sn. This is trivial, as :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} 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==

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