### A Characterization of One-Element p-Bases of Rings of Constants

Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in $K[x{\u2081}^{p},...,x{\u2099}^{p}]$. We prove that $K[x{\u2081}^{p},...,x{\u2099}^{p},f]$ is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.