Consider $\zeta$ a primitive $n$th root of unity, show that

  • if $n\equiv 1\mod{4}\implies \sqrt{n}\in \mathbb{Q}(\zeta)$
  • if $n\equiv -1\mod{4}\implies \sqrt{-n}\in \mathbb{Q}(\zeta)$.

I know that the discriminant for $x^n+a_n$ is $(-1)^{n(n-1)/2}n^na_n^{n-1}$.

i.e. the discriminant for $x^n-1$ will be $n^n$ or $-n^n$ respectively for the two given cases above. We then know that the square root of the discriminant has to lie in $\mathbb{Q}(\zeta)$ - is this correct?

I'm not quite sure how to approach this, should I try to explicitly show that we can get $\sqrt{n}$ and $\sqrt{-n}$ when working in $\mathbb{Q}(\zeta)$ or is there a more abstract way?


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