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I found this statement in Milne's Fields and Galois Theory (page 72):

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More precisely, it is the last line I would like to prove. Could someone help me to prove that question?

I understood everything before the last sentence but now I don't know how to approach this problem.

Any help is appreciated!

Ribbity
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    Take the proof of the equivalent fact for $x^n-a$ and replace the multiplicative hilbert theorem 90 with the additive one and the norm with the trace. Instead of roots of unity, you'll use $\alpha + i$ for $i=1,...,p$ (which is the obstacle to generalizing to higher degree). This result is VI.6.4 in Lang. –  Aug 02 '20 at 21:44
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    Artin–Schreier theory –  Aug 06 '20 at 12:43
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    What Touken Toucan and rain1 said. For what it's worth, I outlined "the book argument" here. Surely the authors of text books have done a better job. – Jyrki Lahtonen Aug 07 '20 at 05:03
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    This is proved here https://stacks.math.columbia.edu/tag/09I7 –  Aug 28 '20 at 20:35

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