2,698 research outputs found
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
Vanishing of eigenspaces and cyclotomic fields
We use a result of Thaine to give an alternative proof of the fact that, for
a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow
subgroup of the ideal class group of \mathbb Q(\zeta_{p}) is trivial.Comment: 6 pages, minor corrections made, to appear in the International
Mathematics Research Notice
Representation of Cyclotomic Fields and Their Subfields
Let \K be a finite extension of a characteristic zero field \F. We say
that the pair of matrices over \F represents \K if \K
\cong \F[A]/ where \F[A] denotes the smallest subalgebra of M_n(\F)
containing and is an ideal in \F[A] generated by . In
particular, is said to represent the field \K if there exists an
irreducible polynomial q(x)\in \F[x] which divides the minimal polynomial of
and \K \cong \F[A]/. In this paper, we identify the smallest
circulant-matrix representation for any subfield of a cyclotomic field.
Furthermore, if is any prime and \K is a subfield of the -th
cyclotomic field, then we obtain a zero-one circulant matrix of size
such that (A,\J) represents \K, where \J is the matrix with
all entries 1. In case, the integer has at most two distinct prime factors,
we find the smallest 0-1 companion-matrix that represents the -th cyclotomic
field. We also find bounds on the size of such companion matrices when has
more than two prime factors.Comment: 17 page
Construction of class fields over cyclotomic fields
Let and be odd primes. For a positive integer let be
the ray class field of modulo . We
present certain class fields of such that , and find the degree of explicitly. And we also
construct, in the sense of Hilbert, primitive generators of the field
over by using Shimura's reciprocity law and special values of theta
constants
Nontrivial Galois module structure of cyclotomic fields
We say a tame Galois field extension with Galois group has trivial
Galois module structure if the rings of integers have the property that
\Cal{O}_{L} is a free \Cal{O}_{K}[G]-module. The work of Greither,
Replogle, Rubin, and Srivastav shows that for each algebraic number field other
than the rational numbers there will exist infinitely many primes so that
for each there is a tame Galois field extension of degree so that has
nontrivial Galois module structure. However, the proof does not directly yield
specific primes for a given algebraic number field For any
cyclotomic field we find an explicit so that there is a tame degree
extension with nontrivial Galois module structure
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