Symmetric and Galois group

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  475821 Galois Group If f(x)=x^7-2 and K is its splitting field over Q, determine the degree of [K:Q], describe K explicitly using a primitive element or by listing a set of generators over Q and find the Galois group of K/Q.

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  Since Hk−1 is normal in G(K,L), L is normal over F, and its Galois group G(L,F) is isomorphic to G(K,F)/Hk−1. It follows that G(L,F) is cyclic.

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  This implies that |Gal(K(alpha)/K)| = [K(alpha):K] = p, whence K(alpha) is Galois with group of order p. Since p is prime, we see that the Galois group must be cyclic. b.

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  Then its Galois group has an element of order l , so by Chebotarev's density theorem there exists p such that the splitting field of f over Fp has Galois group C l , the cyclic group or order l , hence f must be irreducible over Fp