Local Discriminants, Kummerian Extensions, and Elliptic Curves

Starting from Stickelberger’s congruence for the absolute discriminant of a number field, we ask a series of natural questions which ultimately lead to an orthogonality relation for the ramification filtration on K(^p√ K^×), where K is any finite extension of Q_p containing a primitive p-th ischolar_main of 1. An extensive historical survey of discriminants and primary numbers is included. Among other things, we give a direct proof of Serre’s mass formula in the case of quadratic extensions. Incidentally, it is shown that every unit in a local field is the discriminant of some elliptic curve.