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

### Volume 72 / 1995

#### 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 $K₂O_F$ 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.