Abstract: A folklore conjecture asserts that for any fixed prime p and integer n, the size of the p-torsion in the class group of a degree n number field is smaller than any power of the discriminant. In all but a few cases, the best known result towards this conjecture is the trivial bound given by the Brauer-Siegel Theorem.

We make progress on this conjecture by giving the first nontrivial bound on the size of the 2-torsion of the class group of any number field in terms of its discriminant.  Several applications of this result will also be discussed.