Let p and s be prime numbers and let F/Q be an Abelian extension of degree ps or p_. We present formulas for the minimum absolute value of the discriminant of F in terms of p and s. The formulas do not assume the generalized Riemann hypothesis. Instead, they exploit a more general formula for the minimum discriminant based solely on the conductor m of F, on the degree of F, and on the degrees of some particular subfields of F