Journal of Pure and Applied Algebra 166 (2002) 229–238
www.elsevier.com/locate/jpaa
Dyadic ideal, class group, and tame kernel in quadratic
ÿelds
Qin Yue
a; b; ∗
a
Institute of Math., Fudan University, Shanghai 200433, People’s Republic of China
b
Dept. of Math., Xuzhou Normal University Jiangsu 221009, People’s Republic of China
Received 18 August 2000
Communicated by C.A. Weibel
Abstract
For quadratic number ÿelds F with 2 ∈ N
F=Q
(F
∗
), we set up the relations between the class of
dyadic ideal in narrow class group and the element of order 4 in the tame kernel, and determine
the 2
n
(n¿2)-rank of the tame kernel of some quadratic number ÿelds and their Tate kernels.
c
2002 Elsevier Science B.V. All rights reserved.
MSC: 11R65; 11R70; 19C99; 19D50
1. Introduction
Let O
F
be the integral ring of a quadratic ÿeld F =Q(
√
d). Browkin and Schinzel [3]
have given 2-rank formulas and forms of elements of order 2 in K
2
O
F
; Qin [13,14] has
got a method to calculate 4-ranks of K
2
O
F
; Hurrelbrink and Kolster [9] have presented
an eective way of computing 4-ranks of K
2
O
F
for those relative quadratic extensions
via the determination of the F
2
-ranks of certain matrices of local Hilbert symbols.
Recently, for F = Q(
√
d), E = Q(
√
−d); we [7] have given 4-rank formulas of K
2
O
F
:
r
4
(K
2
O
F
)=a(F)+r
4
(C(E));
where a(F)=−1; 0, or 1 is a constant determined eectively by the RedÃei’s matrices
of F and r
4
(C(E)) is 4-rank of the narrow class group C(E)ofE; we [17] have got
the relation between r
4
(K
2
O
F
) and r
4
(R
2
F) (i.e., 4-rank of Hilbert kernel R
2
F).
The paper is supported by Morningside Center of Math. and the Shanghai Postdoctoral Science Foundation.
∗
Corresponding author. Institute of Math., Fudan University, Shanghai 200433, People’s Republic of China.
E-mail address: yueqin2@263.net (Q. Yue).
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c
2002 Elsevier Science B.V. All rights reserved.
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