Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension $L/K$ with Galois group $G$ 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 $l$ so that for each there is a tame Galois field extension of degree $l$ so that $L/K$ has nontrivial Galois module structure. However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For $K$ any cyclotomic field we find an explicit $l$ so that there is a tame degree $l$ extension $L/K$ with nontrivial Galois module structure

LiDIA : a library for computational number theory

In this paper we describe LiDIA, a new library for computational number theory. Why do we work on a new library for computational number theory when such powerful tools as Pari [1], Kant [11], Simath [10] already exist? In fact, those systems are very useful for solving problems for which there exist efficient system routines. For example, using Pari or Kant it is possible to compute invariants of algebraic number fields and Simath can be used to find the rank of an elliptic curve over Q. However, building complicated and efficient software on top of existing systems has in our experience turned out to be very difficult. Therefore, the software of our research group is developed independently of other computer algebra systems

Pseudo-factorials, elliptic functions, and continued fractions

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of "elliptic polynomials", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences.Comment: 24 pages; with correction of typos and minor revision. To appear in The Ramanujan Journa

Symbolic computation: systems and applications

The article presents an overview of symbolic computation systems, their classification-in-history, the most popular CAS, examples of systems and some of their applications. Symbolics versus numeric, enhancement in mathematics, computing nature of CAS, related projects, networks, references are discussed