Quasi-Rational Canonical Forms of a Matrix over a Number Field

Rational Canonical and Jordan Forms

Definition 2.1. Suppose p1, . . . , pk are polynomials in F[x] which are not all 0. Set I = 〈p1, . . . , pk〉. Let d denote the monic generator of I. We call d the greatest common divisor of the pi and write d = gcd(p1, . . . , pk). If d = 1, we say that the polynomials p1, . . . , pk is relatively prime. Note that, by definition, there exists polynomials q1, . . . , qk ∈ F[x] such that d = p1q1...

A scheme over quasi-prime spectrum of modules

The notions of quasi-prime submodules and developed  Zariski topology was introduced by the present authors in cite{ah10}. In this paper we use these notions to define a scheme. For an $R$-module $M$, let $X:={Qin qSpec(M) mid (Q:_R M)inSpec(R)}$. It is proved that $(X, mathcal{O}_X)$ is a locally ringed space. We study the morphism of locally ringed spaces induced by $R$-homomorphism $Mrightar...

Binary Hermitian forms over a cyclotomic field

Article history: Received 21 January 2009 Available online 8 July 2009 Communicated by John Cremona

Nearly Optimal Algorithms for Canonical Matrix Forms

A Las Vegas type probabilistic algorithm is presented for nding the Frobenius canonical form of an n n matrix T over any eld K. The algorithm requires O~(MM(n)) = MM(n) (logn) O(1) operations in K, where O(MM(n)) operations in K are suucient to multiply two n n matrices over K. This nearly matches the lower bound of (MM(n)) operations in K for this problem, and improves on the O(n 4) operations...

Symmetry of Algebras over a Number Field