Computing the tame kernel of quadratic imaginary fields

Authors:
Jerzy Browkin, with an appendix by Karim Belabas and Herbert Gangl

Journal:
Math. Comp. **69** (2000), 1667-1683

MSC (1991):
Primary 19C20; Secondary 11R11, 11R70, 11Y40

Published electronically:
March 15, 2000

MathSciNet review:
1681124

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: J. Tate has determined the group (called the tame kernel) for six quadratic imaginary number fields where Modifying the method of Tate, H. Qin has done the same for and and M. Skaba for and

In the present paper we discuss the methods of Qin and Skaba, and we apply our results to the field

In the Appendix at the end of the paper K. Belabas and H. Gangl present the results of their computation of for some other values of The results agree with the conjectural structure of given in the paper by Browkin and Gangl.

**[A1]**Steven Arno,*The imaginary quadratic fields of class number 4*, Acta Arith.**60**(1992), no. 4, 321–334. MR**1159349****[A2]**Steven Arno, M. L. Robinson, and Ferrell S. Wheeler,*Imaginary quadratic fields with small odd class number*, Acta Arith.**83**(1998), no. 4, 295–330. MR**1610549****[BBCO]**C. Batut, D. Bernardi, H. Cohen, M. Olivier,*GP/PARI Calculator*, version 1.39.**[KH]**K. Belabas, H. Gangl,*Generators and relations for**,**imaginary quadratic*, in preparation.**[BG]**Jerzy Browkin and Herbert Gangl,*Tame and wild kernels of quadratic imaginary number fields*, Math. Comp.**68**(1999), no. 225, 291–305. MR**1604336**, 10.1090/S0025-5718-99-01000-5**[BS]**J. Browkin and A. Schinzel,*On Sylow 2-subgroups of 𝐾₂𝑂_{𝐹} for quadratic number fields 𝐹*, J. Reine Angew. Math.**331**(1982), 104–113. MR**647375**, 10.1515/crll.1982.331.104**[C]**Harvey Cohn,*Advanced number theory*, Dover Publications, Inc., New York, 1980. Reprint of A second course in number theory, 1962; Dover Books on Advanced Mathematics. MR**594936****[Q1]**Hou Rong Qin,*Computation of 𝐾₂𝐙[√-6]*, J. Pure Appl. Algebra**96**(1994), no. 2, 133–146. MR**1303542**, 10.1016/0022-4049(94)90124-4**[Q2]**Hourong Qin,*Computation of 𝐾₂𝑍[(1+√-35)/2]*, Chinese Ann. Math. Ser. B**17**(1996), no. 1, 63–72. A Chinese summary appears in Chinese Ann. Math. Ser. A 17 (1996), no. 1, 121. MR**1387182****[Q3]**-,*Tame kernels and Tate kernels of quadratic number fields*, preprint, 1998.**[S]**Mariusz Skałba,*Generalization of Thue’s theorem and computation of the group 𝐾₂𝑂_{𝐹}*, J. Number Theory**46**(1994), no. 3, 303–322. MR**1273447**, 10.1006/jnth.1994.1016**[T]***Algebraic 𝐾-theory. II: “Classical” algebraic 𝐾-theory, and connections with arithmetic*, Lecture Notes in Mathematics, Vol. 342, Springer-Verlag, Berlin-New York, 1973. Edited by H. Bass. MR**0325308****[W]**Christian Wagner,*Class number 5, 6 and 7*, Math. Comp.**65**(1996), no. 214, 785–800. MR**1333327**, 10.1090/S0025-5718-96-00722-3

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Additional Information

**Jerzy Browkin**

Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL-02-097 Warsaw, Poland

Email:
bro@mimuw.edu.pl

**with an appendix by Karim Belabas**

Affiliation:
Dept. de Mathématiques, Bât. 425, Université Paris-Sud, F-91405 Orsay, France

Email:
Karim.Belabas@math.u-psud.fr

**Herbert Gangl**

Affiliation:
Max-Planck Institut für Mathematik, Vivatsgaße 7, D-53111, Bonn, Germany

Email:
herbert@mpim-bonn.mpg.de

DOI:
http://dx.doi.org/10.1090/S0025-5718-00-01182-0

Keywords:
Tame kernel,
quadratic imaginary fields,
Thue's theorem

Received by editor(s):
January 14, 1998

Received by editor(s) in revised form:
December 7, 1998

Published electronically:
March 15, 2000

Article copyright:
© Copyright 2000
American Mathematical Society