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We give an explicit construction of an integral basis for a radical function field $K=k(t,\rho )$, where ${\rho }^n=D\in k\left[t\right]$, under the assumptions $[K:k(t\left)\right]=n$ and $char\left(k\right)\nmid n$. The field discriminant of $K$ is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the $P$-signatures of a radical function field are also discussed in this paper.