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## Acta Arithmetica

1995 | 72 | 4 | 323-333
Tytuł artykułu

### The 4-rank of $K₂O_F$ for real quadratic fields F

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
1. Introduction. Let F be a number field, and let $O_F$ be the ring of its integers. Several formulas for the 4-rank of $K₂O_F$ are known (see [7], [5], etc.). If √{-1) ∉ F, then such formulas are related to S-ideal class groups of F and F(√(-1)), and the numbers of dyadic places in F and F(√(-1)), where S is the set of infinite dyadic places of F. In [11], the author proposes a method which can be applied to determine the 4-rank of $K₂O_F$ for real quadratic fields F with 2 ∉ NF. The author also lists many real quadratic fields with the 2-Sylow subgroups of $K₂O_F$ being isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/4ℤ. In [12], the author gives a 4-rank $K₂O_F$ formula for imaginary quadratic fields F. By the formula, it is enough to compute some Legendre symbols when one wants to know 4-rank $K₂O_F$ for a given imaginary quadratic field F. In the present paper, we give a similar formula for real quadratic fields F. Then we give 4-rank $K₂O_F$ tables for real quadratic fields F = ℚ√d whose discriminants have at most three odd prime divisors.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
323-333
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-05-24
poprawiono
1994-10-21
Twórcy
autor
• Department of Mathematics, Nanjing University, Nanjing 210008, P.R. China
Bibliografia
• [1] B. Brauckmann, The 2-Sylow subgroup of the tame kernel of number fields, Canad. J. Math. 43 (1991), 215-264.
• [2] J. Browkin and A. Schinzel, On Sylow 2-subgroups of $K₂O_F$ for quadratic fields F, J. Reine Angew. Math. 331 (1982), 104-113.
• [3] A. Candiotti and K. Kramer, On the 2-Sylow subgroup of the Hilbert kernel of K₂ of number fields, Acta Arith. 52 (1989), 49-65.
• [4] P. E. Conner and J. Hurrelbrink, Examples of quadratic number fields with K₂𝓞 containing no element of order four, preprint.
• [5] P. E. Conner and J. Hurrelbrink, The 4-rank of K₂𝓞, Canad. J. Math. 41 (1989), 932-960.
• [6] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York, 1982.
• [7] M. Kolster, The structure of the 2-Sylow subgroup of K₂(O). I, Comment. Math. Helv. 61 (1986), 576-588.
• [8] J. Milnor, Introduction to Algebraic K-theory, Ann. of Math. Stud. 72, Princeton University Press, 1971.
• [9] J. Neukirch, Class Field Theory, Springer, Berlin, 1986.
• [10] O. T. O'Meara, Introduction to Quadratic Forms, Springer, Berlin, 1963.
• [11] H. Qin, The 2-Sylow subgroups of $K₂O_F$ for real quadratic fields F, Science in China Ser. A 23 (12) (1993), 1254-1263 (in Chinese).
• [12] H. Qin, The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields, Acta Arith. 69 (1995), 153-169.
• [13] J. Tate, Relations between K₂ and Galois cohomology, Invent. Math. 36 (1976), 257-274.
• [14] J. Urbanowicz, On the 2-primary part of a conjecture of Birch and Tate, Acta Arith. 43 (1983), 69-81.
Typ dokumentu
Bibliografia
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