Computational number theory

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This collection features a variety of studies and methodologies in the realm of number theory and algebra. It begins with discussions on constructing primitive elements and normal bases in finite fields, followed by numerical methods for determining cyclotomic polynomial coefficients. The exploration of number systems leads to fast-converging series representations of real numbers, with applications in digital processing. The text delves into polynomial transformations and their roles in public key cryptography, as well as number-theoretic transforms linked to a theorem by Sylvester, Kronecker, and Zsigmondy. Algorithms for class groups and regulators are presented, alongside investigations into prime-producing quadratic polynomials and their class numbers. Further contributions include applications of new criteria for class numbers in real quadratic fields and solutions to related problems. The enumeration of quartic fields, computations of class numbers via cyclotomic or elliptic units, and methods for deriving independent units in number fields are also covered. The compilation concludes with insights into Hecke actions on quadratic forms, computations of singular moduli, ranks of elliptic curves, and resolutions of Diophantine equations, including Thue-Mahler equations. Tools like KANT and SIMATH for algebraic number field computations are also introduced, showcasing the interplay between theoretical and practical as