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i know that [tex] F_{q}(x^{\frac{1}{q - 1}}) [/tex] is an extension of [tex] F_{q}(x) [/tex] so i need to find the irreducible polynomial of [tex] x^{\frac{1}{q - 1}} [/tex] over [tex] F_{q}(x)[/tex].

i found this to be [tex] t^{q - 1} - x [/tex] which is irreducible over [tex] F_{q}[x] [/tex] by Eisenstein's criterion. i know that every automorphism in the galois group must map roots of polynomials to roots of the same polynomial but i am having trouble finding the roots of [tex] t^{q - 1} - x [/tex]. besides [tex] x^{\frac{1}{q - 1}} [/tex], im not sure what other roots it could have. can someone give me some hints on this?