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October 24th, 2018, 12:41 PM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 23 Thanks: 0 Math Focus: algebraic number theory  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) $\left(\frac{2,p}{\mathfrak p}\right)=\left(\frac{2}{\mathfrak p}\right)^{v_{\mathfrak p}(p)}= \left( \frac{2}{\mathfrak p} \right)=(1)^{\frac{p^21}{8}}=1$ 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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