Related papers: Families of Cyclic Cubic Fields
We produce an explicit family of totally real cyclic quartic polynomials that are monogenic in many cases and, if the $abc$ conjecture holds, generate distinct monogenic quartic fields infinitely often. Additional families (also…
We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.
We define standardized constructions of finite fields, and standardized generators of (multiplicative) cyclic subgroups in these fields. The motivation is to provide a substitute for Conway polynomials which can be used by various software…
We introduce an orbifold induction procedure which provides a systematic construction of cyclic orbifolds, including their twisted sectors. The procedure gives counterparts in the orbifold theory of all the current-algebraic constructions…
This paper presents an algebraic construction of families of unitary matrices that achieve full diversity. They are obtained as subsets of cyclic division algebras.
We show how to construct a cubic partial cube from any simplicial arrangement of lines or pseudolines in the projective plane. As a consequence, we find nine new infinite families of cubic partial cubes as well as many sporadic examples.
We construct an infinite family of real cyclotomic fields with non-trivial class group. This result generalizes the result in [1] in the sense that our family includes theirs.
We classify all cubic extensions of any field of arbitrary characteristic, up to isomorphism, via an explicit construction involving three fundamental types of cubic forms. We deduce a classification of any Galois cubic extension of a…
In addition to rather complicated general methods it is interesting and valuable to develop fast efficient methods for calculating generators of power integral bases in special types of number fields. We consider sextic fields containing a…
Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…
We show how to construct infinite families of explicitly determined cubic number fields whose class group has a subgroup isomorphic to $(\mathbb{Z}/2)^8$ using degree $1$ del Pezzo surfaces. We illustrate the method and provide an example…
In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic…
For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in…
Algebraic methods are used to construct families of unramified abelian extensions of some families of number fields with specified Galois groups.
Using a technique \cite{holgernorma2002} to construct a basis for spinors and ``families'' of spinors in terms of Clifford algebra objects, we define other Clifford algebra objects, which transform the state of one ''family'' of spinors…
We give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters.
In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…
Class field theory furnishes an intrinsic description of the abelian extensions of a number field that is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such…
In previous work we determined automorphism groups of cyclic algebraic curves defined over fields of any odd characteristic. In this paper we determine parametric equations of families of curves for each automorphism group for such curves.
Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family.…