On elementary abelian 2-Sylow K₂ of rings of integers of certain quadratic number fields

A large number of papers have contributed to determining the structure of the tame kernel ${\displaystyle K{₂}_F}$ of algebraic number fields F. Recently, for quadratic number fields F whose discriminants have at most three odd prime divisors, 4-rank formulas for ${\displaystyle K{₂}_F}$ have been made very explicit by Qin Hourong in terms of the indefinite quadratic form x² - 2y² (see [7], [8]). We have made a successful effort, for quadratic number fields F = ℚ (√(±p₁p₂)), to characterize in terms of positive definite binary quadratic forms, when the 2-Sylow subgroup of the tame kernel of F is elementary abelian. This makes determining exactly when the 4-rank of ${\displaystyle K{₂}_F}$ is zero, computationally even more accessible. For arbitrary algebraic number fields F with 4-rank of ${\displaystyle K{₂}_F}$ equal to zero, it has been pointed out that the Leopoldt conjecture for the prime 2 is valid for F, compare [6]. We consider this paper to be an addendum to the Acta Arithmetica publications [7], [8]. It grew out of our circulated 1989 notes [3]. Acknowledgements. We gratefully acknowledge fruitful long-term communications on this topic with Jerzy Browkin.