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Norm of $2^{i}$-th primitive root
Let $K=\mathbb{Q}[\sqrt{-2}]$. Then $\frac{1}{2}\sqrt{-2}(1-i)$ is an eighth root of unity in $L$ whose norm is $\frac{1}{2}\sqrt{-2}(1-i)\frac{1}{2}\sqrt{-2}(1+i)=\frac{1}{4}(-2)(1-i^2)=-1\in K$. [I'...
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