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#類体論へ至る道
$\mathbb{Q}({}^4\sqrt{2},i)$の元$x$が
$x=a+b({}^4\sqrt{2})+c({}^4\sqrt{2})^2+d({}^4\sqrt{2})^3$
$+ei+f({}^4\sqrt{2})i+g({}^4\sqrt{2})^2i+h({}^4\sqrt{2})^3i$
$(a〜h\in \mathbb{Q})$
…①

$τ$を作用すると
$τ(x)=a+b({}^4\sqrt{2})+c({}^4\sqrt{2})^2+d({}^4\sqrt{2})^3$
$-ei-f({}^4\sqrt{2})i-g({}^4\sqrt{2})^2i-h({}^4\sqrt{2})^3i$
…②
で$τ(x)=x$になるには①=②なので
　$e=f=g=h=0$
よって①の元$x$は
　$x=a+b({}^4\sqrt{2})+c({}^4\sqrt{2})^2+d({}^4\sqrt{2})^3$
…③