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Elements of small norm in Shanks' cubic extensions of imaginary quadratic fields

2004
*
Journal of symbolic computation
*

be an

doi:10.1016/j.jsc.2004.05.005
fatcat:o63artypzbb7njobcqei7gr6c4
*imaginary**quadratic*number*field*with ring*of*integers Z k and let k(α) be the*cubic**extension**of*k generated by the polynomial*In*the present paper we characterize all*elements*γ ∈ Z k [α] with ... This generalizes a corresponding result by Lemmermeyer and Pethő for*Shanks*'*cubic**fields*over the rationals. ... Acknowledgement The authors were supported by project S8310*of*the Austrian Science Foundation. ...##
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Gaussian periods and units in certain cyclic fields

1992
*
Proceedings of the American Mathematical Society
*

We analyze the property

doi:10.1090/s0002-9939-1992-1093600-5
fatcat:ygn46fmyirehpffztoeazz2xm4
*of*period-unit integer translation (there exists a Gaussian period n and rational integer c such that n + c is a unit)*in*simplest*quadratic*,*cubic*, and quartic*fields**of*arbitrary ... This is an*extension**of*work*of*E. Lehmer, R. Schoof, and L. C. Washington for prime conductor. We also determine the Gaussian period polynomial for arbitrary conductor. ... Introduction*In*[10] , Lehmer exhibited a remarkable property*of*the simplest*cubic**fields**of**Shanks*[13] and the simplest quartic*fields**of*Gras [4] . ...##
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Gaussian Periods and Units in Certain Cyclic Fields

1992
*
Proceedings of the American Mathematical Society
*

We analyze the property

doi:10.2307/2159341
fatcat:tmkvwfdgobhsfi2smyzu7v7vx4
*of*period-unit integer translation (there exists a Gaussian period n and rational integer c such that n + c is a unit)*in*simplest*quadratic*,*cubic*, and quartic*fields**of*arbitrary ... This is an*extension**of*work*of*E. Lehmer, R. Schoof, and L. C. Washington for prime conductor. We also determine the Gaussian period polynomial for arbitrary conductor. ... Introduction*In*[10] , Lehmer exhibited a remarkable property*of*the simplest*cubic**fields**of**Shanks*[13] and the simplest quartic*fields**of*Gras [4] . ...##
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Page 3972 of Mathematical Reviews Vol. , Issue 96g
[page]

1996
*
Mathematical Reviews
*

The authors show that certain integers do not occur as the

*norms**of*principal ideals*in*the “simplest*cubic**fields*”*of*D.*Shanks*. ... Cohen observed*in*1989 that the statement seems to hold for*fields**of**small*degree. ...##
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Heuristics on class groups
[chapter]

1984
*
Lecture notes in mathematics
*

They are äs followsi (A) If p is a

doi:10.1007/bfb0071539
fatcat:z5kosp2wlncnzasdl3gik4sroe
*small*odd prime, the proportion*of**imaginary**quadratic**fields*whose class number h(o) is divisible by p is significantly greater than l/p. ... Buell [1] for iraaginary*quadratic**fields*, and*Shanks*and Williams [5] for real guadcatic*fields*. ... If p is a*small*odd prime, the proportion*of**imaginary**quadratic**fields*whose class number h(o) is divisible by p is significantly greater than l/p. ...##
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Class numbers of cyclotomic fields

1985
*
Journal of Number Theory
*

ACKNOWLEDGMENT The authors would like to thank Daniel

doi:10.1016/0022-314x(85)90055-1
fatcat:i7fa3zph4raanfqfae2q4zmqcu
*Shanks*for his generous assistance during the preparation*of*this paper. ...*In*the above we used*cubic*(or sextic) and quartic*fields*corresponding to*elements**of*finite order*in*PGL,(Q). ... The "simplest*cubic**fields*"*of*Daniel*Shanks*[ 121 provide a suitable family*of**cubic**fields*. Let a be an integer and let p be a root*of*x'--aP-(a+3)X-1. ...##
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Page 5292 of Mathematical Reviews Vol. , Issue 91J
[page]

1991
*
Mathematical Reviews
*

*Shanks*studied the simplest

*cubic*

*fields*[Math. Comp. 28 (1974), 1137-1152; MR 50 #4537], and M.-N. ... Childs (1-SUNYA) 91j:11094 11R23 Sands, Jonathan W. (1-VT) On

*small*Iwasawa invariants and

*imaginary*

*quadratic*

*fields*. Proc. Amer. Math. Soc. 112 (1991), no. 3, 671-684. ...

##
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Construction of hyperelliptic function fields of high three-rank

2008
*
Mathematics of Computation
*

Most

doi:10.1090/s0025-5718-07-02001-7
fatcat:a4sikn73pbgz7amwizlqq4f7cy
*of*our methods are adapted from analogous techniques used for generating*quadratic*number*fields*whose ideal class groups have high 3-rank, but one method, applicable to finding large l-ranks for ... Our focus is on finding examples for which the genus and the base*field*are as*small*as possible. ...*In*the first two cases, K/F q (x) is an*imaginary**quadratic**extension*, whilst*in*the latter scenario, K/F q (x) is real. ...##
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Page 3 of Mathematical Reviews Vol. , Issue 89H
[page]

1989
*
Mathematical Reviews
*

Let K be an

*imaginary**quadratic**field*. ... (The reviewer has considered the case*of**quadratics*and*cubics*[Proc. Amer. Math. Soc. 7 (1956), 595-598; MR 18 #114].) The literature for degree 3, 4, 5, and 6 includes works by D.*Shanks*[Math. ...##
###
Reduced Ideals in Pure Cubic Fields
[article]

2019
*
arXiv
*
pre-print

*In*the case

*of*pure

*cubic*

*fields*, generated by cube roots

*of*integers, a convenient integral basis provides a means for identifying reduced ideals

*in*these

*fields*. ... Reduced ideals have been defined

*in*the context

*of*integer rings

*in*

*quadratic*number

*fields*, and they are closely tied to the continued fraction algorithm. ... Introduction

*Quadratic*

*fields*have been studied much more

*extensively*than their

*cubic*analogues. ...

##
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The value of L(12, χ) for Abelian L-functions of complex quadratic fields

1984
*
Journal of Number Theory
*

From this relation a criterion is derived for the vanishing at s = f

doi:10.1016/0022-314x(84)90062-3
fatcat:fljiffril5dobfmknhehdjs7n4
*of*the Dedekind zeta function for the non-cyclic*cubic**extensions**of*the rational*field*. ... It is shown that the values*of*Abelian L-functions*of*complex*quadratic**fields*at s = f can be expressed as finite sums*of*values*of*a non-holomorphic modular form at certain special points*in*the Poincare ...*In*his paper on pure*cubic**fields*[3] Dedekind made an*extensive*study*of*the class groups C, for the Eisenstein*field*k = Q((-3)"2) using the law*of**cubic*reciprocity. ...##
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Infrastructure: Structure Inside the Class Group of a Real Quadratic Field

2014
*
Notices of the American Mathematical Society
*

Acknowledgment The authors thank Sarah Chisholm, Monireh Rezai Rad and Peter Zvengrowski for their careful proofreading and suggestions for improvement

doi:10.1090/noti1064
fatcat:libv27koarcunbwyosenj7uxe4
*of*this article. ...*In*the 1980s*Shanks*began to compute class numbers*of**imaginary**quadratic**fields*with his programmable hand-held calculator. ...*Shanks*originally formulated his NUCOMP method for*imaginary**quadratic**fields*, but it was extended to real*quadratic**fields*by van der Poorten [30] . ...##
###
Index to Volumes 37 and 38

2004
*
Journal of symbolic computation
*

ideal

doi:10.1016/s0747-7171(04)00109-9
fatcat:q3cckydpknhjhinygacsvlj52y
*of*a general projective curve, 295 BELABAS, K., A relative van Hoeij algorithm over number*fields*, 641 BERNSTEIN, D., The computational complexity*of*rules for the character table*of*S n , 727 BURCKEL ... , 641 A symbolic test for (i, j )-uniformity*in*reduced zero-dimensional schemes, 403 AHN, M. ... over*small*finite*fields*, 1227 JOSWIG, M. and ZIEGLER, G.M., Convex hulls, oracles, and homology, 1247 KIRSCHENHOFER, P. and THUSWALDNER, J.M.,*Elements**of**small**norm**in**Shanks*'*cubic**extensions**of**imaginary*...##
###
Heuristics on class groups of number fields
[chapter]

1984
*
Lecture notes in mathematics
*

*in*particular

*imaginary*and real

*quadratic*

*fields*. ... B/ If P is a

*small*odd prime, the proportion

*of*

*imaginary*

*quadratic*

*fields*whose class number is divisible by p seems to be significantly greater than l/p (for instance 43% for p = 3, 23.5% for p=5). ...

*Shanks*, L. Washington, D. Zagier for valuable discussions during the preparation

*of*this paper and D. Buell, C. P. Schnorr, D.

*Shanks*and H. ...

##
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HT90 and "simplest" number fields
[article]

2012
*
arXiv
*
pre-print

Taking n=3, we recover Shanks's simplest

arXiv:1207.6099v1
fatcat:grvmtg3sybacfnzqeat6sanlzq
*cubic**fields*. ... The "simplest" number*fields**of*degrees 3 to 6, Washington's cyclic quartic*fields*, and a certain family*of*totally real cyclic*extensions**of*(cos(π/4m)) all have defining polynomials whose zeroes satisfy ... Derek Holt's description*of*8 T 11 as a central product, and Laurent Bartholdi's use*of*GAP were instrumental*in*understanding 8 T 11 and 20 T 53 . H. W. Lenstra, Jr. and ...
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