On the existence of ramified abelian covers

Given a normal complete variety $Y$ over an algebraically closed field $\mathbb K$, distinct effective Weil divisors $D_1,... D_n$ of $Y$ and positive integers $d_1,... d_n$, we spell out the conditions for the existence of an abelian cover of $Y$ branched with order $d_i$ on $D_i$. As an application, we prove that a cover of a normal complete toric variety branched on the torus-invariant divisors is itself a toric variety if the characteristic of $\mathbb K$ is equal to 0 or if the cover is Galois of degree not divisible by the characteristic.