On the strongly ambiguous classes of some biquadratic number fields

Abstract

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\Bbbk=\Bbb Q(\sqrt{2pq}, {\rm i})$, where ${\rm i}=\sqrt{-1}$ and $p\equiv-q\equiv1 \pmod4$ are different primes. For each of the three quadratic extensions $\Bbb K/\Bbbk$ inside the absolute genus field $\Bbbk^{(*)}$ of $\Bbbk$, we determine a fundamental system of units and then compute the capitulation kernel of $\Bbb K/\Bbbk$. The generators of the groups ${\rm Am}_s(\Bbbk/F)$ and ${\rm Am}(\Bbbk/F)$ are also determined from which we deduce that $\Bbbk^{(*)}$ is smaller than the relative genus field $(\Bbbk/\Bbb Q({\rm i}))^*$. Then we prove that each strongly ambiguous class of $\Bbbk/\Bbb Q({\rm i})$ capitulates already in $\Bbbk^{(*)}$, which gives an example generalizing a theorem of Furuya (1977).

Keywords

• absolute genus field
• relative genus field
• fundamental system of units
• 2-class group
• capitulation
• quadratic field
• biquadratic field
• multiquadratic CM-field