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
Primitive roots of unity and field automorphisms of equal order
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abstract-algebra
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0Sorry 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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0Thanks 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