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how can I characterize the minimal polynomials of all elements of this extension 'to show that they have all the distinct roots?

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    $\begingroup$ You can show more generally that, if $E$ is an extension field of a field $K$ of characteristic $0$, then $E$ is a separable extension. This is because $\gcd\big(p(x),p'(x)\big)=1$ for all irreducible polynomial $p(x)\in K[x]$. This is something that may fail in positive characteristics. $\endgroup$ – Batominovski Nov 12 '18 at 21:29
  • $\begingroup$ This is a duplicate. $\endgroup$ – Dietrich Burde Nov 12 '18 at 21:39
  • $\begingroup$ @DietrichBurde, it's not a direct duplicate of the question in the body. $\endgroup$ – lhf Nov 12 '18 at 23:11
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More generally, we have this:

Every irreducible polynomial over a field of characteristic zero has distinct roots.

Indeed, $f'\ne0$ and $\deg f' < \deg f$ imply $\gcd(f,f')=1$.

(The part that can fail in positive characteristic is $f'\ne0$.)

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