Harbingers of Artin's Reciprocity Law. IV. Bernstein's Reciprocity Law

In the last article of this series we will first explain how Artin's reciprocity law for unramified abelian extensions can be formulated with the help of power residue symbols, and then show that, in this case, Artin's reciprocity law was already stated by Bernstein in the case where the base field contains the roots of unity necessary for realizing the Hilbert class field as a Kummer extension. Bernstein's article appeared in 1904, almost 20 years before Artin conjectured his version of the reciprocity law, and seems to have been overlooked completely

Harbingers of Artin's Reciprocity Law. III. Gauss's Lemma and Artin's Transfer

We briefly review Artin's reciprocity law in the classical ideal theoretic language, and then study connections between Artin's reciprocity law and the proofs of the quadratic reciprocity law using Gauss's Lemma

Ideal class groups of cyclotomic number fields II

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions

Families of noncongruent numbers

Let E_k denote the elliptic curve defined by y^2 = x(x^2 - k^2). We consider the curves with k = pl, p = l = 1 mod 8 primes, and show that the density of rank-0 curves among them is at least 1/2 by explicitly constructing nontrivial elements in the 2-part of the Tate-Shafarevich group of E_k