A recent note proves that Okuyama’s characterization of associated primes in group cohomology does not need the base field to be algebraically closed, as previously thought. The result applies to every field of characteristic p that divides the order of the group. The proof replaces the earlier dependence on a lemma that required an algebraically closed field with a construction based on Lemma 6.3. Lemma 6.3 builds a linear invariant in a symmetric algebra under a finite group action, circumventing the earlier reliance on a result for p‑groups and a contested proof by Benson. The argument uses the Normal Basis Theorem to produce an element whose orbit under the group is linearly independent, a key step that extends the theorem’s validity to arbitrary fields.

This advance broadens the applicability of Okuyama’s theorem, which links associated primes of the cohomology ring of a finite group to elementary abelian p‑subgroups. By removing the algebraically closed requirement, the theorem now applies in more general algebraic contexts while preserving its original conclusions.