Inductive methods for counting number fields

Abstract We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle’s Conjecture and counterexamples to Malle’s Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group 𝐺. Our method relies on having asymptotic counts for 𝑇-extensions for some normal subgroup 𝑇 of 𝐺, uniform bounds for the number of such 𝑇-extensions, and possibly weak bounds on the asymptotic number of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>T</m:mi> </m:mrow> </m:math> G/T -extensions. However, we do not require that most 𝑇-extensions of a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>T</m:mi> </m:mrow> </m:math> G/T -extension are 𝐺-extensions. Our new results use 𝑇 either abelian or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>S</m:mi> <m:mn>3</m:mn> <m:mi>m</m:mi> </m:msubsup> </m:math> S_{3}^{m} , though our framework is general.