oblixps
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i am trying to find G(F_{q}(x^{\frac{1}{q - 1}}/F_{q}(x)) where q is the power of some prime.
i know that F_{q}(x^{\frac{1}{q - 1}}) is an extension of F_{q}(x) so i need to find the irreducible polynomial of x^{\frac{1}{q - 1}} over F_{q}(x).
i found this to be t^{q - 1} - x which is irreducible over F_{q}[x] 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 t^{q - 1} - x. besides x^{\frac{1}{q - 1}}, I am not sure what other roots it could have. can someone give me some hints on this?
i know that F_{q}(x^{\frac{1}{q - 1}}) is an extension of F_{q}(x) so i need to find the irreducible polynomial of x^{\frac{1}{q - 1}} over F_{q}(x).
i found this to be t^{q - 1} - x which is irreducible over F_{q}[x] 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 t^{q - 1} - x. besides x^{\frac{1}{q - 1}}, I am not sure what other roots it could have. can someone give me some hints on this?