Computation of class numbers of quadratic number fields
Author:
Stéphane Louboutin
Journal:
Math. Comp. 71 (2002), 17351743
MSC (2000):
Primary 11R11, 11R29, 11R21, 11Y35
Published electronically:
November 21, 2001
MathSciNet review:
1933052
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.
 [Coh]
Henri
Cohen, A course in computational algebraic number theory,
Graduate Texts in Mathematics, vol. 138, SpringerVerlag, Berlin,
1993. MR
1228206 (94i:11105)
 [Dav]
Harold
Davenport, Multiplicative number theory, 2nd ed., Graduate
Texts in Mathematics, vol. 74, SpringerVerlag, New York, 1980.
Revised by Hugh L. Montgomery. MR 606931
(82m:10001)
 [Lou1]
Stéphane
Louboutin, 𝐿functions and class numbers
of imaginary quadratic fields and of quadratic extensions of an imaginary
quadratic field, Math. Comp.
59 (1992), no. 199, 213–230. MR 1134735
(92k:11128), http://dx.doi.org/10.1090/S00255718199211347356
 [Lou2]
Stéphane
Louboutin, Computation of relative class numbers
of CMfields, Math. Comp.
66 (1997), no. 219, 1185–1194. MR 1422790
(97k:11157), http://dx.doi.org/10.1090/S0025571897008636
 [Lou3]
Stéphane
Louboutin, Computation of relative class numbers of imaginary
abelian number fields, Experiment. Math. 7 (1998),
no. 4, 293–303. MR 1678103
(2000c:11207)
 [Lou4]
Stéphane
Louboutin, Computation of relative class numbers
of CMfields by using Hecke 𝐿functions, Math. Comp. 69 (2000), no. 229, 371–393. MR 1648395
(2000i:11172), http://dx.doi.org/10.1090/S0025571899010960
 [Lou5]
S. Louboutin.
Computation of and of relative class numbers of CMfields. Nagoya Math. J. 161 (2001), 171191. CMP 2001:09
 [MoWi]
R.
A. Mollin and H.
C. Williams, Computation of the class number of a real quadratic
field, Utilitas Math. 41 (1992), 259–308. MR 1162532
(93d:11134)
 [ScWa]
René
Schoof and Lawrence
C. Washington, Quintic polynomials and real
cyclotomic fields with large class numbers, Math. Comp. 50 (1988), no. 182, 543–556. MR 929552
(89h:11067b), http://dx.doi.org/10.1090/S00255718198809295522
 [StWi]
A.
J. Stephens and H.
C. Williams, Computation of real quadratic fields
with class number one, Math. Comp.
51 (1988), no. 184, 809–824. MR 958644
(90b:11106), http://dx.doi.org/10.1090/S00255718198809586447
 [WiBr]
H.
C. Williams and J.
Broere, A computational technique for
evaluating 𝐿(1,𝜒) and the class number of a real quadratic
field, Math. Comp. 30
(1976), no. 136, 887–893. MR 0414522
(54 #2623), http://dx.doi.org/10.1090/S00255718197604145225
 [Coh]
 H. Cohen.
A Course in Computational Algebraic Number Theory. SpringerVerlag, Grad. Texts Math. 138, 1993. MR 94i:11105
 [Dav]
 H. Davenport.
Multiplicative Number Theory, The functional Equation of the Functions. SpringerVerlag, Grad. Texts Math. 74 (1980), Chapter 9. MR 82m:10001
 [Lou1]
 S. Louboutin.
functions and class numbers of imaginary quadratic fields and of quadratic extensions of an imaginary quadratic field. Math. Comp. 59 (1992), 213230. MR 92k:11128
 [Lou2]
 S. Louboutin.
Computation of relative class numbers of CMfields. Math. Comp. 66 (1997), 173184. MR 97k:11157
 [Lou3]
 S. Louboutin.
Computation of relative class numbers of imaginary abelian number fields. Experimental Math. 7 (1998), 293303. MR 2000c:11207
 [Lou4]
 S. Louboutin.
Computation of relative class numbers of CMfields by using Hecke functions. Math. Comp. 69 (2000), 371393. MR 2000i:11172
 [Lou5]
 S. Louboutin.
Computation of and of relative class numbers of CMfields. Nagoya Math. J. 161 (2001), 171191. CMP 2001:09
 [MoWi]
 R. A. Mollin and H. C. Williams.
Computation of the class number of a real quadratic field. Utilitas Math. 41 (1992), 259308. MR 93d:11134
 [ScWa]
 R. Schoof and L. C. Washington.
Quintic polynomials and real cyclotomic fields with large class numbers. Math. Comp. 50 (1988), 543556. MR 89h:11067b
 [StWi]
 A. J. Stephens and H. C. Williams.
Computation of real quadratic fields with class number one. Math. Comp. 51 (1988), 809824. MR 90b:11106
 [WiBr]
 H. C. Williams and J. Broere.
A computational technique for evaluating and the class number of a real quadratic field. Math. Comp. 30 (1976), 887893. MR 54:2623
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (2000):
11R11,
11R29,
11R21,
11Y35
Retrieve articles in all journals
with MSC (2000):
11R11,
11R29,
11R21,
11Y35
Additional Information
Stéphane Louboutin
Affiliation:
Institut de Mathématiques de Luminy, UPR 906, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
Email:
loubouti@iml.univmrs.fr
DOI:
http://dx.doi.org/10.1090/S0025571801013679
PII:
S 00255718(01)013679
Keywords:
Quadratic number field,
class number,
Dirichlet $L$function,
relative class number.
Received by editor(s):
March 29, 2000
Received by editor(s) in revised form:
November 27, 2000
Published electronically:
November 21, 2001
Article copyright:
© Copyright 2001 American Mathematical Society
