Number Field Counting Conjecture (Malle’s Conjecture)

Establish, for every number field k and every transitive permutation group G of degree n, an asymptotic formula for the number of degree-n extensions L/k whose Galois closure over k has Galois group isomorphic to G (as a permutation group), ordered by absolute discriminant, of the form #F_{n,k}(G;X) ~ c X^{1/a} (log X)^{b-1} as X → ∞ for some positive constants a, b, c depending on k and G.

Background

Counting number fields by discriminant is a central problem in arithmetic statistics. For a number field k and a transitive permutation group G of degree n, the quantity #F_{n,k}(G;X) counts degree-n extensions L/k whose Galois closure has permutation group G, with absolute discriminant bounded by X.

Malle formulated a general prediction for the growth of #F_{n,k}(G;X), proposing explicit exponents a(G) and b(k,G) and a constant c depending on k and G. This conjecture unifies and extends many previously known cases and guides modern approaches to counting number fields. The present paper proves many new cases and counterexamples to the predicted b in broad families, but the conjecture remains open in general.