Explicit discriminant of a class of polynomial, monogenity and Galois group

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Let K = Q ( θ ) where θ is a root of an irreducible polynomial f ( x ) = x n − km ( x k + a ) m + b ∈ Z [ x ] , 1 ≤ km < n and Z K denote the ring of algebraic integers of K. In this paper, we explicitly compute the discriminant of f ( x ) and provide necessary and sufficient conditions involving only a, b, m, k, n for f ( x ) to be monogenic. Moreover, we characterize all the primes dividing the index of the subgroup Z [ θ ] in Z K . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group S n , the permutation group on n letters.