98 research outputs found
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
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