We present a new algorithm for constructing a local group of prime powers. We show that the new algorithm vastly outperforms existing ones. As a side product, we also obtain an algorithm for checking whether a local field contains a primitive root of unity of prime degree and a new method for testing if an element is a prime power in a completion of a number field.

The article presents the package for the computer algebra system Magma . It is a collection of functions for manipulating quaternionic polynomials over number fields. It provides nearly the same functionality for quaternionic polynomials as the ones provided by Magma for polynomials over fields.

We propose a novel method for constructing a basis of the group of local square-classes of a dyadic completion of a number field. The described method has been implemented by the author in the computer algebra systems Magma and SageMath. It turned out to be substantially faster than the algorithm previously used. The article presents the algorithm itself, together with a time comparison with the preexisting solution.

Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for which there are known numerical methods for polynomial root approximation. The sole purpose of the present paper is to present a polynomial factorization algorithm for division quaternion algebras over number fields, together with its adaptation for root finding.

Pourchet proved in 1971 that every nonnegative univariate polynomial with rational coefficients is a sum of five or fewer squares. Nonetheless, there are no known algorithms for constructing such a decomposition. The sole purpose of the present paper is to present a set of algorithms that decompose a given nonnegative polynomial into a sum of six (five under some unproven conjecture or when allowing weights) squares of polynomials. Moreover, we prove that the binary complexity can be expressed polynomially in terms of classical operations of computer algebra and algorithmic number theory.

The problem of writing a totally positive element as a sum of squares has a long history in mathematics, going back to Bachet and Lagrange. While for some specific rings (like integers or polynomials over the rationals), there are known methods for decomposing an element into a sum of squares, in general, for many other important rings and fields, the problem is still widely open. In this paper, we present an explicit algorithm for decomposing an element of an arbitrary global field (either a number field or a global function field) into a sum of squares of minimal length.

We present two new algorithms for computing an anisotropic part of a non-degenerate quadratic form over a global function field of odd characteristic.

The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtwängler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and S-singular elements (modulo squares) in a global function field.

We show a method for constructing a polynomial interpolating roots’ multiplicities of another polynomial, that does not use companion matrices. This leads to a modification to Guersenzvaig–Szechtman square-free decomposition algorithm that is more efficient both in theory and in practice.