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Let $K$ be a field, and $\alpha$ be an element of a separable extension of $K$, such that $\alpha^p\in K$, but $\alpha\notin K$, $p$ a prime.

Let $f$ be the minimal polynomial of $\alpha$ over $K$ and $\beta$ be another root of $f$.

Question 1. Is it true that $\alpha$ and $\beta$ differ by a $p$-th root of unity? i.e. if $\epsilon=\alpha/\beta$, the is $\epsilon^p=1$?

Question 2. In general, is it true that any two roots of an irreducible polynomial over a field always differ (multiplicatively) by a root of unity?


Reference: Galois Theory - Ian Stewart, Third Ed., p. 156.

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(1) Let $f(x)$ be the minimal polynomial of $\alpha$. Let $k=\alpha^p$ (note $k\in K$). Since $\alpha^p-k=0$, we must have $f(x)\mid (x^p-k)$, and so $\beta$ is also a root of $x^p-k$, thus $\alpha^p=k=\beta^p~\Rightarrow (\alpha/\beta)^p=1$.

(2) No. There's no reason for them to in general. The needn't be the same size even. For instance, over $\Bbb Q$, the roots of the irreducible quadratic $(x-1)^2-2$ don't have the same size.

In general, the only relations satisfied by roots of a polynomial are Vieta's formulas. (There probably exists some kind of clever formal way to state this idea.)

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