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Finding irreducible polynomials over finite fields
Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe
Sequences of irreducible polynomials without prescribed coefficients over odd prime fields
In this paper we construct infinite sequences of monic irreducible
polynomials with coefficients in odd prime fields by means of a transformation
introduced by Cohen in 1992. We make no assumptions on the coefficients of the
first polynomial of the sequence, which belongs to \F_p [x], for some
odd prime , and has positive degree . If for
some odd integer and non-negative integer , then, after an initial
segment with , the degree of the polynomial
is twice the degree of for any .Comment: 10 pages. Fixed a typo in the reference
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
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