We show that under the assumption of a certain Generalized Riemann Hypothesis the problem of verifying the value of the class number of an arbitrary algebraic number eld F of arbitrary degree belongs to the complexity class NP \ co ?NP. In order to prove this result we introduce a compact representation of algebraic integers which allows us to represent a system of fundamental units by (2 + log...

The algorithm for factoring polynomials over the integers by Wang and Rothschild is generalized to an algorithm for the irreducible factorization of multivariate polynomials over any given algebraic number field. The extended method makes use of recent ideas in factoring univariate polynomials over large finite fields due to Berlekamp and Zassenhaus. The procedure described has been implemented...

Grosu (2016) [11] asked if there exist an integer r≥3 and a finite family of r-graphs whose Turán density, as real number, has (algebraic) degree greater than r−1. In this note we show that, for all integers d, exists density at least thus answering Grosu's question in strong form.

Motivated by symbolic dynamics, we study the problem, given a unital subring 5 of the reals, when is a matrix A algebraically shift equivalent over S to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of A are sufficient, and establish the conjecture in many cases. If S is the integers, we give some lower bounds on sizes of realizing primitive matrices...

We present a probabilistic algorithm for testing the result of the product of two n-bit integers (polynomials of degree n over any field) in 4n+o (n) bit (algebraic) operations with the error probability 0 (n-£j for any e < 0.5. .The first version of this paper was written at the Hebrew University of Jerusalem and presented at 1981 ACM Symposium on Algebraic and Symbolic Computation.

Fermat famously claimed in the margin of a book that a certain family of Diophantine equations have no solutions in integers. For over 300 years Fermat’s claim remained unsolved, and it provided motivation for many important developments in algebraic number theory. We will develop some of the foundational ideas of modern algebraic number theory in the context of Fermat’s Last Theorem, and sketc...