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Let $K/k$ a field extension, $[K:k]=p$ prime, $f(X)\in [X]$ with degree $\deg(f)= p+1$, so $f$ has root in $K\iff f$ has root in $k$.

As $k\subset K$, the $\Leftarrow$ implication is trivial. Any hints to the $\Rightarrow$ implication? I really don't know what to do.

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Here's one Hint:

If $\xi\in K$ is a root of $f$, consider the subextension $$k\subset k(\xi)\subset K$$ and remember the degree of extensions is multiplicative.

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  • $\begingroup$ Ok, so $p=[K:k]=[K:k(\alpha)][k(\alpha):k]$. Since $p$ is prime, $[K:k(\alpha)]=p$ or $[k(\alpha):k]=p$. If $[K:k(\alpha)]=p$, $[k(\alpha):k]=1\implies \alpha\in k$. But what if $[k(\alpha):k]=p$? $\endgroup$ – Mateus Rocha Oct 9 at 18:40
  • $\begingroup$ @BrianMoehring: Silly me! I messed with all those multiplicativity, divisibility, &c. I'll delete it in a moment. $\endgroup$ – Bernard Oct 9 at 18:59
  • $\begingroup$ @BrianMoehring, can you help me with my comment, please? I would really appreciate that $\endgroup$ – Mateus Rocha Oct 10 at 3:46

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