### $*$-extremal valued fields.

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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.