## A Course in Computational Algebraic Number TheoryOne of the first of a new generation of books in mathematics that show the reader how to do large or complex computations using the power of computer algebra. It contains descriptions of 148 algorithms, which are fundamental for number theoretic calculations, in particular for computations related to algebraic number theory, elliptic curves, primality testing, lattices and factoring. For each subject there is a complete theoretical introduction. A detailed description of each algorithm is given allowing for immediate computer implementation. Many of the algorithms are new or appear for the first time in this book. A large number of exercises is also included. |

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### Contents

Fundamental NumberTheoretic Algorithms | 1 |

Algorithms for Linear Algebra and Lattices | 45 |

Algorithms on Polynomials | 108 |

Copyright | |

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### Common terms and phrases

algebraic integer algebraic number algorithm computes assume Chapter class group class number coefficients congruence coprime corresponding cubic fields defined definition degree denote discriminant dividing division divisor element elliptic curve equal equation equivalent Euclidean step example Exercise exists factor base finite following algorithm formula fractional ideal function fundamental discriminant Galois group give given go to step hence Hermite normal form Initialize integral basis inverse irreducible polynomial isomorphic kernel lattice Legendre symbol Lemma linear LLL algorithm matrix method minimal polynomial modulo monic multi-precision multiplication non-trivial non-zero norm notations Note number field number theory obtain Otherwise output prime ideals prime number primitive problem Proof Proposition prove quadratic fields quadratic form real numbers reduced form resp result roots of unity Section sieve square squarefree Sub-algorithm subfield terminate the algorithm Theorem trivial unique Z-basis Z-module