English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that$\displaystyle\left(\frac{2}{p_1}\right)=\displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$.Put $d=p_1p_2q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field${k}=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group,$\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalizationof the $2$-classes of ${k}$ in its fourteen unramified abelian extensions$\mathbb{K}_j$ and $\mathbb{L}_j$ within ${k}_2^{(1)}$, that is the Hilbert$2$-class field of ${k}$. We determine the nilpotency class, the coclass,generators and the structure of the metabelian Galois group$G=\mathrm{Gal}(\mathbb{L}/{k})$ of the second Hilbert 2-class field${k}_2^{(2)}$ of ${k}$. Additionally, the abelian type invariants of the groups$\mathbf{C}l_2(\mathbb{K}_j)$ and $\mathbf{C}l_2(\mathbb{L}_j)$ and the lengthof the $2$-class tower of ${k}$ are given.

Principalization of $2$-class groups of type $(2,2,2)$ of biquadratic fields $\mathbb{Q}\left(\sqrt{\strut p_1p_2q},\sqrt{\strut -1}\right)$