منابع مشابه
Visualizing imaginary quadratic fields
Imaginary quadratic fields Q( √ −d), for integers d > 0, are perhaps the simplest number fields afterQ. They are equal parts helpful first example and misleading special case. LikeZ, the Gaussian integersZ[i] (the cased = 1) have unique factorization and a Euclidean algorithm. As d grows, however, these properties eventually fail, first the latter and then the former. The classical Euclidean al...
متن کاملFast ideal cubing in imaginary quadratic number and function fields
We present algorithms for computing the cube of an ideal in an imaginary quadratic number field or function field. In addition to a version that computes a non-reduced output, we present a variation based on Shanks’ NUCOMP algorithm that computes a reduced output and keeps the sizes of the intermediate operands small. Extensive numerical results are included demonstrating that in many cases our...
متن کاملReal and imaginary quadratic representations of hyperelliptic function fields
A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor’...
متن کاملEuclidean Ideals in Quadratic Imaginary Fields
— We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique class that generates the class-group is moreover norm-Euclidean.
متن کاملVisualising the arithmetic of quadratic imaginary fields
We study the orbit of R under the Bianchi group PSL2(OK), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK , is a geometric realisation, as an intricate circle packing, of the arithmetic of K. This paper presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples of √ −∆ and describe the curvatures of t...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2010
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2009.09.014