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

Peter Kirschenhofer, Jörg M. Thuswaldner
2004 Journal of symbolic computation  
be an 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.  ... 
doi:10.1016/j.jsc.2004.05.005 fatcat:o63artypzbb7njobcqei7gr6c4

Gaussian periods and units in certain cyclic fields

Andrew J. Lazarus
1992 Proceedings of the American Mathematical Society  
We analyze the property 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] .  ... 
doi:10.1090/s0002-9939-1992-1093600-5 fatcat:ygn46fmyirehpffztoeazz2xm4

Gaussian Periods and Units in Certain Cyclic Fields

Andrew J. Lazarus
1992 Proceedings of the American Mathematical Society  
We analyze the property 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] .  ... 
doi:10.2307/2159341 fatcat:tmkvwfdgobhsfi2smyzu7v7vx4

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 fieldsof D. Shanks.  ...  Cohen observed in 1989 that the statement seems to hold for fields of small degree.  ... 

Heuristics on class groups [chapter]

H. Cohen, H. W. Lenstra
1984 Lecture notes in mathematics  
They are äs followsi (A) 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.  ...  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.  ... 
doi:10.1007/bfb0071539 fatcat:z5kosp2wlncnzasdl3gik4sroe

Class numbers of cyclotomic fields

Gary Cornell, Lawrence C Washington
1985 Journal of Number Theory  
ACKNOWLEDGMENT The authors would like to thank Daniel 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.  ... 
doi:10.1016/0022-314x(85)90055-1 fatcat:i7fa3zph4raanfqfae2q4zmqcu

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.  ... 

Construction of hyperelliptic function fields of high three-rank

M. Bauer, M. J. Jacobson, Y. Lee, R. Scheidler
2008 Mathematics of Computation  
Most 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.  ... 
doi:10.1090/s0025-5718-07-02001-7 fatcat:a4sikn73pbgz7amwizlqq4f7cy

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]

George Jacobs
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.  ... 
arXiv:1905.00242v3 fatcat:5wnry2mq55brlkl7sit4bqq5mm

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

Carlos J Moreno
1984 Journal of Number Theory  
From this relation a criterion is derived for the vanishing at s = f 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.  ... 
doi:10.1016/0022-314x(84)90062-3 fatcat:fljiffril5dobfmknhehdjs7n4

Infrastructure: Structure Inside the Class Group of a Real Quadratic Field

Michael J. Jacobson, R. Scheidler
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 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] .  ... 
doi:10.1090/noti1064 fatcat:libv27koarcunbwyosenj7uxe4

Index to Volumes 37 and 38

2004 Journal of symbolic computation  
ideal 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  ... 
doi:10.1016/s0747-7171(04)00109-9 fatcat:q3cckydpknhjhinygacsvlj52y

Heuristics on class groups of number fields [chapter]

H. Cohen, H. W. Lenstra
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.  ... 
doi:10.1007/bfb0099440 fatcat:y24tszjiwfcmldpkyohjzvjtvi

HT90 and "simplest" number fields [article]

Kurt Foster
2012 arXiv   pre-print
Taking n=3, we recover Shanks's simplest 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  ... 
arXiv:1207.6099v1 fatcat:grvmtg3sybacfnzqeat6sanlzq
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