Cohen and Lenstra have given a heuristic which, for a fixed odd prime p, leads to many interesting predictions about the distribution of p-class groups of imaginary quadratic fields. We extend the Cohen-Lenstra heuristic to a non-abelian setting by considering, for each imaginary quadratic field K, the Galois group of the p-class tower of K, i.e.

Abstract. We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture thatis a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when r is large with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r ≤ 8, and (iii) when n ≤ 4. The main tool is the theory of p-adic Newton Polygons.

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R 0 (2m) be the minimal root discriminant for totally complex number fields of degree 2m, and put α 0 = lim infm R 0 (2m). Define R 1 (m) to be the minimal root discriminant of totally real number fields of degree m and put α 1 = lim infm R 1 (m). Assuming the Generalized Riemann Hypothesis, α 0 ≥ 8πe γ ≈ 44.7, and, α 1 ≥ 8πe γ+π/2 ≈ 215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates for α 0 and α 1 : α 0 < 82.2, α 1 < 954.3.

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