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Let $\Phi_n$ be the nth cyclotomic polynomial.

I would like to show that if $4$ divides $n$, then $\Phi_n$ is even. Any idea ?

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Two ideas:

  • if $n=4k$, then $\phi(n)=\deg(\Phi_n(x))$ is even
  • $z$ is a $4k$-th primitive root if and only if $\;-z\;$ is

Now write the cyclotomic polynomial as product of linear factors and conclude

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