The 4-rank of $K_2O_F$ for real quadratic fields F

Hourong Qin

ArticleOriginal scientific text

Title

The 4-rank of $K_{2} O_{F}$ for real quadratic fields F

Authors ¹

Affiliations

Department of Mathematics, Nanjing University, Nanjing 210008, P.R. China

Abstract

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

K ₂ O_{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

K_{2} O_{F}

for real quadratic fields F with 2 ∉ NF. The author also lists many real quadratic fields with the 2-Sylow subgroups of

K_{2} O_{F}

being isomorphic to ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/4ℤ. In [12], the author gives a 4-rank

K_{2} O_{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

K_{2} O_{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

K_{2} O_{F}

tables for real quadratic fields F = ℚ√d whose discriminants have at most three odd prime divisors.

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Title

The 4-rank of K2OF for real quadratic fields F

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Abstract

Bibliography

The 4-rank of $K_{2} O_{F}$ for real quadratic fields F