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 Chems October 24th, 2018 12:41 PM

Where is the problem?? Norm residue symbol

Hello
Let $p$ be a rational prime such that $p\equiv 3\mod 8$ and let $\mathfrak p$ the unique prime ideal of $\mathbb{Q}(i)$ laying over $p$.
Applying the norm residue properties i get :

1) $\left(\frac{2,p}{\mathfrak p}\right)=\left(\frac{2}{\mathfrak p}\right)^{v_{\mathfrak p}(p)}= \left( \frac{2}{\mathfrak p} \right)=(-1)^{\frac{p^2-1}{8}}=-1$

2) $\left(\frac{2,p}{\mathfrak p}\right)=\left(\frac{N_{\mathbb Q(i)/\mathbb Q} (2),p}{ p}\right)=\left(\frac{4,p}{ p}\right)=1$.

!! what is the mistake??

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