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-   -   Where is the problem?? Norm residue symbol (http://mymathforum.com/number-theory/345198-where-problem-norm-residue-symbol.html)

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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