Discriminant and integral basis of number fields defined by exponential Taylor polynomials

Let Kₙ=Q(αₙ) be a family of algebraic number fields where αₙ∈ C is a root of the n th exponential Taylor polynomial xⁿ/n!+ xⁿ⁻¹(n-1)!+ ⋯ +x²/2!+x/1!+1 , n∈ N . In this paper, we give a formula for the exact power of any prime p dividing the discriminant of K n in terms of the p -adic expansion of n . An explicit p -integral basis of K n is also given for each prime p . These p -integral bases quickly lead to the construction of an integral basis of K n .