Today I have read that the order of the splitting field of $x^d+c$ always divide $d\phi(d)$. It looks like a nice result, but my text does not give any reference for the proof. Could anyone suggest me some text where to study it?
2026-03-25 13:13:26.1774444406
The order of the splitting field of $x^d+c$ always divide $d\phi(d)$
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You might find some help in any treatment of cyclic extensions or Kummer extensions, but quite honestly with a little basic field theory you can probably get there yourself. The main ingredient is looking at the roots of $x^d+c$ assuming the ground field contains the $d^{th}$ roots of unity, hence the divides $d$, and the assumption about containing the $d^{th}$ roots of unity accounts for the $\varphi (d)$.