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We give a constructive proof of Gilmer’s theorem that if every nonzero polynomial over a field k has a root in some fixed extension field E, then each polynomial in k [X] splits in E [X]. Using a slight generalization of this theorem, we construct, in a functorial way, a commutative, discrete, von Neumann regular k-algebra A so that each polynomial in k [X] splits in A [X].

A theorem of Gilmer and the canonical universal splitting ring