1,887 research outputs found
Splitting full matrix algebras over algebraic number fields
Let K be an algebraic number field of degree d and discriminant D over Q. Let
A be an associative algebra over K given by structure constants such that A is
isomorphic to the algebra M_n(K) of n by n matrices over K for some positive
integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with
M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm
is a deterministic procedure which is allowed to call oracles for factoring
integers and factoring univariate polynomials over finite fields.)
As a consequence, we obtain a polynomial time ff-algorithm to compute
isomorphisms of central simple algebras of bounded degree over K.Comment: 15 pages; Theorem 2 and Lemma 8 correcte
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases
This paper is a sequel to "Computing diagonal form and Jacobson normal form
of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We
present a new fraction-free algorithm for the computation of a diagonal form of
a matrix over a certain non-commutative Euclidean domain over a computable
field with the help of Gr\"obner bases. This algorithm is formulated in a
general constructive framework of non-commutative Ore localizations of
-algebras (OLGAs). We split the computation of a normal form of a matrix
into the diagonalization and the normalization processes. Both of them can be
made fraction-free. For a matrix over an OLGA we provide a diagonalization
algorithm to compute and with fraction-free entries such that
holds and is diagonal. The fraction-free approach gives us more information
on the system of linear functional equations and its solutions, than the
classical setup of an operator algebra with rational functions coefficients. In
particular, one can handle distributional solutions together with, say,
meromorphic ones. We investigate Ore localizations of common operator algebras
over and use them in the unimodularity analysis of transformation
matrices . In turn, this allows to lift the isomorphism of modules over an
OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of
this lifting with the solutions of the original system of equations. Moreover,
we prove some new results concerning normal forms of matrices over non-simple
domains. Our implementation in the computer algebra system {\sc
Singular:Plural} follows the fraction-free strategy and shows impressive
performance, compared with methods which directly use fractions. Since we
experience moderate swell of coefficients and obtain simple transformation
matrices, the method we propose is well suited for solving nontrivial practical
problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio
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