Related papers: Computing square roots in quaternion algebras

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…

For an odd prime $p$, we say a polynomial $f\in \mathbb F_p[X]$ computes square roots if $f(a)^2=a$ for all nonzero, perfect squares $a\in \mathbb F_p$. When $p\equiv 3 \mod 4$, it is easy to see that $f(X)=X^{\frac{p+1}{4}}$ is the…

In this paper we determine sufficient conditions for a quaternion algebra to split over a quadratic field. In the last section of the paper, we find a class of division symbol algebras of degree $n$ (where $n$ is a positive integer, $n\geq…

This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.

We present an explicit basis for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms.

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for…

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2x2-matrix…

In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise…

{\small In this paper, we find a class of division quaternion algebras over the field }$\mathbb{Q}\left( i\right) ${\small \ and a class of division symbol algebras over a cyclotomic field.}

We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of…

Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the…

The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.

We show how to construct unramified qoaternion extensions of quadratic number fields.

We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…

We classify the four dimensional perfect non-simple evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.

Let $k$ be a field of characteristic $\neq 2$. In this paper we study squares, cubes and their products in split and anisotropic groups of type $A_1$. In split case, we show that computing $n^{\rm th}$ roots is equivalent to finding…

Our contribution is two-folded. First, starting from the known fact that every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give a complete characterization of the square roots of a real skew-Hamiltonian matrix W.…

We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.

We discuss existence of factorizations with linear factors for (left) polynomials over certain associative real involutive algebras, most notably over Clifford algebras. Because of their relevance to kinematics and mechanism science, we put…

This short article presents explicit expressions for roots of a quartic equation that has all four real roots. Although a general expression for quartic roots is available on Wikipedia, an optimized and slightly shorter expression for only…