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Whilst reading P.M. Cohn's "Skew Field Constructions" (LMS LNS 27; Cambridge 1977), I found myself unable to follow the author's argument on p. 69 (proof of Prop. 3.5.4, specifically lines 15-16). Most of my troubles would disappear if I could prove the following: given an automorphism $\sigma$ of a (commutative) field $K$ of order $n$ and a primitive $n$-th root of unity $\omega\in K$ (i.e. an element of order $n$ in the multiplicative group of $K$), where $1, one necessarily has $\sigma(\omega)=\omega$. Now I'm not even sure whether this statement is true, though a look at Ex. § 11, 4) c), Chapter V of Bourbaki's Algebra II makes me hopeful. Any help, one way or the other, would be greatly appreciated !

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    Sorry about that Shaun, I've corrected the question; naturally $\sigma$ should have order $n$. (A scan of the page in question wouldn't really be of any help, because my question doesn't appear there as such - an answer, however, would help in making the remaining arguments comprehensible. The reference was given simply to provide context.)2012-09-05

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I think this is a counterexample. Let $K={\bf Q}(\zeta)$, where $\zeta$ is a primitive 20th root of 1. Define $\sigma$ by $\sigma(\zeta)=\zeta^3$. Then $\sigma$ is of order 4, and $\sigma(i)=-i$.

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    Thanks very much Gerry - looks good to me ! (And unfortunately means that I still don't understand Cohn's argument in his book - but that is another question ...) - Kind regards !2012-09-05