Wir betrachten die ${\displaystyle {}\mathbb {Z} /(2)\times \mathbb {Z} /(2)\times \mathbb {Z} /(2)}$-graduierte Körpererweiterung

${\displaystyle {}\mathbb {Q} \subseteq \mathbb {Q} [{\sqrt {3}},{\sqrt {5}},{\sqrt {7}}]\,}$

und den durch ${\displaystyle {}\chi (e_{1})=-1}$, ${\displaystyle {}\chi (e_{2})=1}$, ${\displaystyle {}\chi (e_{3})=-1}$, gegebenen Charakter. Bestimme

${\displaystyle \varphi _{\chi }{\left(4+5{\sqrt {3}}-3{\sqrt {5}}+6{\sqrt {7}}+7{\sqrt {15}}-2{\sqrt {21}}+13{\sqrt {35}}-4{\sqrt {105}}\right)}.}$