On the construction of division chains in algebraic number rings,with applications to SL<sub>2</sub>

Cooke, George E.; Weinberger, P.

doi:10.1080/00927877508822057

Cited by 47 publications

(26 citation statements)

References 5 publications

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“…On GRH, an analogue of [3] [11] was only proven recently by Roskam ([15]). We want to extend the result of Narkiewicz for inert primes.…”

Section: 1mentioning

confidence: 99%

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Primitive Roots in Quadratic Fields

Cohen

2006

Int. J. Number Theory

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Abstract. We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture.

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“…On GRH, an analogue of [3] [11] was only proven recently by Roskam ([15]). We want to extend the result of Narkiewicz for inert primes.…”

Section: 1mentioning

confidence: 99%

“…Assuming GRH, Cooke and Weinberger ( [3]) and Lenstra ([11]) showed that given a real quadratic field K, there are infinitely many split primes for which the fundamental unit of the field 1 has maximal order (namely p − 1) in C ǫ (p).…”

Section: 1mentioning

confidence: 99%

Primitive Roots in Quadratic Fields

Cohen

2006

Int. J. Number Theory

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“…In this paper we are concerned with a generalized form of Artin's conjecture, which recently arose in connection with Euclid's algorithm [23,30,19] and the construction of division chains [5,20] in global fields.…”

Section: Jentroductiopmentioning

confidence: 99%

“…all p e S split completely in Fs and moreover F is the largest field with these propn.rties, inside an algebraic closure of K; cf. [5].…”

Section: Examplesmentioning

confidence: 99%

On Artin's conjecture and Euclid's algorithm in global fields

1977

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Abstract.-This paper considers a generalization of Artin's conjecture on primes with prescribed primitive roots. The main result provides & necessary and sufficient condition for the conjectural density of certain sets of primes to be non-zero. As an application a theorem about the existence of a euclidean algorithm in rings of arithtnetic type is proved.Key_word£: Artin's conjecture, primitive roots, Euclid's algorithm.12A75, 10HI5, 12A45, 13F10.

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“…Under a generalized Riemann hypothesis, Weinberger [32] shows that every unique factorization domain Od, d > 0, is Euclidean with respect to some degree function, and in Cooke and Weinberger [7] it is shown that a constant bound for the number of divisions can be achieved, namely 5. In fact, these results are shown for the rings of algebraic integers in arbitrary algebraic number fields, provided there are infinitely many units.…”

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confidence: 99%

Computing greatest common divisors and factorizations in quadratic number fields

Kaltofen¹,

Rolletschek²

1989

Math. Comp.

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Abstract.In a quadratic number field Q(V~D), D a squarefree integer, with class number 1, any algebraic integer can be decomposed uniquely into primes, but for only 21 domains Euclidean algorithms are known. It was shown by Cohn [5] that for D < -19 even remainder sequences with possibly nondecreasing norms cannot determine the GCD of arbitrary inputs. We extend this result by showing that there does not even exist an input in these domains for which the GCD computation becomes possible by allowing nondecreasing norms or remainders whose norms are not as small as possible. We then provide two algorithms for computing the GCD of algebraic integers in quadratic number fields Q(s/D). The first applies only to complex quadratic number fields with class number 1, and is based on a short vector construction in a lattice. Its complexity is 0(S3), where S is the number of bits needed to encode the input. The second algorithm allows us to compute GCD's of algebraic integers in arbitrary number fields (ideal GCD's if the class number is > 1). It requires only 0(S2) binary steps for fixed D, but works poorly if D is large. Finally, we prove that in any domain, the computation of the prime factorization of an algebraic integer can be reduced in polynomial time to the problem of factoring its norm into rational primes. Our reduction is based on a constructive version of a theorem by A. Thue.

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On the construction of division chains in algebraic number rings,with applications to SL₂

Cited by 47 publications

References 5 publications

Primitive Roots in Quadratic Fields

Primitive Roots in Quadratic Fields

On Artin's conjecture and Euclid's algorithm in global fields

Computing greatest common divisors and factorizations in quadratic number fields

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On the construction of division chains in algebraic number rings,with applications to SL2

Cited by 47 publications

References 5 publications

Primitive Roots in Quadratic Fields

Primitive Roots in Quadratic Fields

On Artin's conjecture and Euclid's algorithm in global fields

Computing greatest common divisors and factorizations in quadratic number fields

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Product

Resources

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On the construction of division chains in algebraic number rings,with applications to SL₂