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A Course in Computational Algebraic Number Theory
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TLDR
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods.Abstract:
A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.read more
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References
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Journal ArticleDOI
Approximating rings of integers in number fields
TL;DR: This paper studies the algorithmic problem of finding the ring of integers of a given algebraic mimber field and proves that this subring has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields.
Journal ArticleDOI
A Monte Carlo factoring algorithm with linear storage
TL;DR: An algorithm which will factor an integer n quite efficiently if the class number h(-n) is free of large prime divisors and the method requires an amount of storage space which is proportional to the length of the input n.
Journal ArticleDOI
Computing Galois groups over the rationals
Leonard H. Soicher,John McKay +1 more
TL;DR: In this paper, practical computational techniques are described to determine the Galois group of a polynomial over the rationals, and each transitive permutation group of degree 3 to 7 is realized as a Galois groups over rationals.
Journal ArticleDOI
Algorithms for Hermite and Smith normal matrices and linear Diophantine equations
TL;DR: A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables that is valid if the elements of the matrix are in a principal ideal domain.
Journal ArticleDOI
On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
TL;DR: In this article, the authors verified the WeilTaniyama conjecture for the elliptic curve y2 = 4x3 28x + 25 with a triple zero at s = 1 and computed lim, L(s)/(s 1)3 to 28 decimal places, in accordance with the Birch-Swinnerton-Dyer conjecture.
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