The discriminant $d_F$ of a number field $F$ is a basic invariant of $F$. The smaller $d_F$ is, the more elements there are in the ring of integers $O_F$ of $F$ that have a given bounded size. This is relevant, for example, to cryptography using elements of $O_F$. In 2007, two cryptographers (Peikert and Rosen) asked whether one could give an explicit construction of an infinite family of number fields $F$ having $d_F^{1/[F:Q]}$ smaller than $[F:Q]^d$ for some $d < 1$. By an explicit construction we mean an algorithm requiring time bounded by a polynomial in $log([F:Q])$ for producing a set of polynomials whose roots generate $F$. In this talk I will describe work with Ted Chinburg showing how this can be done for any $d > 0$. The proof uses the group theory of profinite 2-groups as well as recent results in analytic number theory.
