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If $p \equiv 1 \pmod n$, what additional conditions are needed to ensure that $$2^{(p-1)/n} \equiv 1 \pmod p?$$

I know:

  • For $n=3$ (cubic reciprocity) the form is $p=x^2+27y^2$.
  • For $n=4$ (biquadratic reciprocity) the form is $p=x^2+64y^2$.
  • The solution has to deal with Gaussian integers and Artin's reciprocity theorem.

Can someone please show me how exactly to do this, maybe with the example $n=5$.

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  • $\begingroup$ I found it very hard to understand the question, so I re-phrased it, hopefully without changing the meaning. Feel free to revert or re-edit if I did not succeed. I think you also mean to assume that $p$ is prime. I also mention your earlier question mathoverflow.net/questions/362187/… , which was the case $n = 3$. $\endgroup$
    – LSpice
    Commented Jun 4, 2020 at 17:04
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    $\begingroup$ If $n$ is a prime and $\mathbb Q(\zeta_n)$ has class number 1, then you can use Eisenstein reciprocity to give a similar criterion. $\endgroup$
    – Wojowu
    Commented Jun 4, 2020 at 17:11
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    $\begingroup$ I strongly suggest that you read the excellent and very complete book of Franz Lemmermeyer (who is on MO) on the subject. $\endgroup$
    – Henri Cohen
    Commented Jun 4, 2020 at 22:22

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