In this paper, I explain the Quadratic Sieve, its Multiple Polynomial variation, the Number Field Sieve, and give some worked examples of the afore-mentioned algorithms. Using my own Maple implementation of the Quadratic Sieve, I explore the effect of altering one of the parameters of the Quadratic Sieve algorithm, with respect to both time and success rate.

In a paper of Kim, Chan, and Rhagavan, the universal ternary classical quadratic forms over quadratic fields of positive discriminant were discovered. Here a proof of the universality of some of these quadratic forms is given using a technique of Liouville. Another quadratic form over the field of discriminant 8 is shown universal by a different elementary approach. Mathematics Subject Classifi...

A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor’...

A new technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely ‘algebraic’ method. They are obtained by means of specialising certain modular equations related to Weber’s modular equations of ‘irrational type’. The technique works for a large class of et...

We complete the task, begun in [19], of determining when the central norm (determined by the infrastructure of the underlying real quadratic field) is equal to 2 in the simple continued fraction expansion of the associated quadratic surd.

We study 2-incompressible Grassmannians of isotropic subspaces of a quadratic form, of a hermitian form over a quadratic extension of the base field, and of a hermitian form over a quaternion algebra.

This article provides necessary and sufficient conditions for a real quadratic field to have class number one or two in terms of a new set of primeproducing quadratic polynomials