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38
Computing Arakelov class groups
, 2008
"... Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of ..."
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Cited by 7 (0 self)
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Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context.
Algorithms for Quadratic Orders
 PROCEEDINGS OF SYMPOSIUM ON MATHEMATICS OF COMPUTATION
, 1993
"... We describe deterministic algorithms for solving the following algorithmic problems in quadratic orders: Computing fundamental unit and regulator, principal ideal testing, solving prime norm equations, computing the structure of the class group, computing the order of an ideal class and determining ..."
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Cited by 5 (2 self)
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We describe deterministic algorithms for solving the following algorithmic problems in quadratic orders: Computing fundamental unit and regulator, principal ideal testing, solving prime norm equations, computing the structure of the class group, computing the order of an ideal class and determining discrete logarithms in the class group. We also prove upper bounds for the time and space complexity of the algorithms.
Solvability by Radicals from an Algorithmic Point of View
, 2001
"... Any textbook on Galois theory contains a proof that a polynomial equation with solvable Galois group can be solved by radicals. From a practical point of view, we need to nd suitable representations of the group and the roots of the polynomial. We first reduce the problem to that of cyclic extension ..."
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Cited by 4 (1 self)
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Any textbook on Galois theory contains a proof that a polynomial equation with solvable Galois group can be solved by radicals. From a practical point of view, we need to nd suitable representations of the group and the roots of the polynomial. We first reduce the problem to that of cyclic extensions of prime degree and then work out the radicals, using the work of Girstmair. We give numerical examples of Abelian and nonAbelian solvable equations and apply the general framework to the construction of Hilbert Class fields of imaginary quadratic fields.
A Bound for the Torsion in the KTheory of Algebraic Integers
 DOCUMENTA MATH.
, 2003
"... Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the mth Quillen Kgroup of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discrimin ..."
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Cited by 3 (0 self)
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Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the mth Quillen Kgroup of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant.
Identifying the Matrix Ring: ALGORITHMS FOR QUATERNION ALGEBRAS AND QUADRATIC FORMS
, 2010
"... We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We d ..."
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We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.
Complex multiplication tests for elliptic curves
, 2004
"... Abstract. We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminan ..."
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Cited by 2 (0 self)
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Abstract. We consider the problem of checking whether an elliptic curve defined over a given number field has complex multiplication. We study two polynomial time algorithms for this problem, one randomized and the other deterministic. The randomized algorithm can be adapted to yield the discriminant of the endomorphism ring of the curve.
ON PERFECT POWERS IN LUCAS SEQUENCES
"... Abstract. Let (un)n≥0 be the binary recurrent sequence of integers given by u0 = 0, u1 = 1 and un+2 = 2(un+1 + un). We show that the only positive perfect powers in this sequence are u1 = 1 and u4 = 16. We also discuss the problem of determining perfect powers in Lucas sequences in general. ..."
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Cited by 2 (1 self)
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Abstract. Let (un)n≥0 be the binary recurrent sequence of integers given by u0 = 0, u1 = 1 and un+2 = 2(un+1 + un). We show that the only positive perfect powers in this sequence are u1 = 1 and u4 = 16. We also discuss the problem of determining perfect powers in Lucas sequences in general.