After a review of some central objects in algebraic number theory, we will discuss a certain pairing on p-units of the cyclotomic field F of p-th roots of unity, for an irregular prime p. This pairing takes values in essentially the class group of F modulo p. McCallum and I have conjectured that it is surjective, which I have now proven for p < 1000. We will discuss a variety of applications of this pairing: to the structure of class groups, to the K-theory of integer rings, to a Lie algebra associated to the geometric fundamental group of the projective line minus three points, and to cusp forms congruent to Eisenstein series modulo p.