We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res(P,Pi) of the minimal polynomial P of a primitive integral element and of its irreducible factors Pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the ca...

We present new, e cient algorithms for computations on separable matrix algebras over in nite elds. We provide a probabilistic of the Monte Carlo type to nd a generator for the centre of a given algebra A Fm m over an in nite eld F using a number of operations that is within a logarithmic factor of the cost of solving m m systems of linear equations. A Las Vegas algorithm is also provided under...

We investigate the complexity of 1) computing the characteristic polynomial, the minimal polynomial, and all the invariant factors of an integer matrix, and of 2) verifying them, when the coefficients are given as input. It is known that each coefficient of the characteristic polynomial of a matrix A is computable in GapL, and the constant term, the determinant of A, is complete for GapL. We sh...

In this paper, two classes of generalized cyclotomic sequences of period pq are reconstructed by means of multi-rate parallel combinations of binary Legendre sequences which are clocked at different rates. Then these generalized cyclotomic sequences can be generated by combinations of short and cheap LFSR’s. From the multi-rate constructions and the trace representation of binary Legendre seque...

The aim of the paper is to compile and compare basic theoretical facts on Krylov subspaces and block Krylov subspaces. Many Krylov (sub)space methods for solving a linear system Ax = b have the property that in exact computer arithmetic the true solution is found after ν iterations, where ν is the dimension of the largest Krylov subspace generated by A from r0, the residual of the initial appro...

We present new, eecient algorithms for computations on separable matrix algebras over innnite elds. We provide a probabilistic of the Monte Carlo type to nd a generator for the centre of a given algebra A F mm over an innnite eld F using a number of operations that is within a logarithmic factor of the cost of solving m m systems of linear equations. A Las Vegas algorithm is also provided under...

For a positive integer m, set ζm = exp(2πi/m) and let Z ∗ m denote the group of reduced residues modulo m. Fix a congruence group H of conductor m and of order f . Choose integers t1, . . . , te to represent the e = φ(m)/f cosets of H in Zm. The Gauss periods θj = ∑ x∈H ζ tjx m (1 ≤ j ≤ e) corresponding toH are conjugate and distinct over Q with minimal polynomial g(x) = x + c1x e−1 + · · ·+ ce...

1.1. Field extensions. Suppose K is a field containing a subfield F . An element α ∈ K is called algebraic over F if there exists a non-zero polynomial f ∈ F [x] such that f(α) = 0. If α is algebraic over F , the set Iα(F ) := {f ∈ F [x] | f(α) = 0} is a non-zero ideal of F [x] (this is easy, but do check it). Since F [x] is a PID (it is in fact a Euclidean ring with respect to the degree funct...

In this paper, we consider some vector extrapolation methods for solving nonsymmetric systems of linear equations. When applied to sequences generated linearly, these methods namely the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE), are Krylov subspaces methods and are respectively equivalent to the method of Arnoldi and to the GCR and GMRES. By considering the...

In this paper, we introduce a C∗-algebra associated with a proper primitive substitution. We show that the C∗-algebra is simple and purely infinite and contains the associated CuntzKrieger algebra and the crossed product C∗-algebra of the corresponding Cantor minimal system. And we calculate the K-groups.