On the class number of real multiquadratic fields

Abstract

In their 1988 monograph, Conner and Hurrelbrink classified all real quadratic and biquadratic fields with odd class number. They also investigated real quadratic fields whose $2$-class numbers are equal to $2$. Their results, together with some well-known results of Ku{č}era, provide a complete classification of real quadratic number fields with $2$-class number $2$. The main aim of this paper is to provide a complete classification of all real biquadratic fields whose $2$-class numbers equal $2$; and moreover, we draw the complete list all real multiquadratic fields with odd class numbers. Finally, as an appendix we give explicit formulas for $\sqrt{\varepsilon_d}$, where $\varepsilon_d$ denotes the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$.