Ok so i have the spliting field of $f(x)=x^{p}-3$, say it's $F$, and i know $[F:\mathbb Q]= p(p-1)$ I have to show there exists only one extension $L$ such $[F:L]=p-1$, that there is one It comes directly from the spliting field, but How one can prove it's the only one?
2026-03-27 11:46:32.1774611992
Existence of only one extension
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Hint: show that the Galois group of $F$ has a unique subgroup of order $p-1$ (it is a semi-direct product of two cyclic groups) and apply the fundamental theorem of the Galois theory.