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I have been thinking about the field traces of roots of unity. Let $\zeta$ be a primitive $n$-th root of unity, and let $K$ be any subfield of $\mathbb{Q}[n]$. If I take the trace of $\zeta$ down to $K$ and set $x := Tr_K^{\mathbb{Q}[n]}(\zeta)$, is it possible that $x$ could lie in a proper subfield of $K$, or will it always be true that $K = \mathbb{Q}(x)$.

If this were true, I'm guessing a proof of this might involve the fact that $\mbox{Gal}(\mathbb{Q}[n])$ acts regularly on the primitive $n$-th roots of unity, and also the fact that $\mathbb{Q}[n]$ is a Galois extension of $\mathbb{Q}$. I can't seem to answer this myself though. I don't have an especially strong background in algebraic number theory.

Chris
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    It is very much possible that you end up with something in a smaller field. I have used the following example. Let $n=8$. Let $K$ be the fixed field of the automorphism $\zeta\mapsto \zeta^5$. Because $\zeta^5=-\zeta$ you get $tr(\zeta)=0$. – Jyrki Lahtonen Apr 02 '24 at 11:49
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    See here for more about linear dependencies among the primitive roots of unity. My example with $n=8$ is, after all, a manifestation of that. – Jyrki Lahtonen Apr 02 '24 at 16:43

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