24,067 research outputs found
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
On the Hidden Shifted Power Problem
We consider the problem of recovering a hidden element of a finite field
\F_q of elements from queries to an oracle that for a given x\in \F_q
returns for a given divisor . We use some techniques from
additive combinatorics and analytic number theory that lead to more efficient
algorithms than the naive interpolation algorithm, for example, they use
substantially fewer queries to the oracle.Comment: Moubariz Garaev (who has now become a co-author) has introduced some
new ideas that have led to stronger results. Several imprecision of the
previous version have been corrected to
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