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The following result may help to determine the Galois group of an irreducible polynomial.

Theorem : Let $G$ be a transitive subgroup of the symmetric group $S_n$ and p a prime number in $]\frac n2;n-2[$. If $G$ contains a p-cycle then G is either $S_n$ or $A_n$.

Does somebody have a comprehensive proof of this fact ? Using basic arguments, please. References are wellcome.

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Regarding a reference, Coleman uses this exact result in his paper about Galois groups of exponential polynomials. I haven't seen the proof yet, but he cites Marshall Hall's 1959 edition of 'The Theory of Groups', Theorems 5.6.2 and 5.7.2.

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