A theorem of Gilmer and the canonical universal splitting ring (Q2453704)

In his paper [Am. Math. Mon. 75, 1101--1102 (1968; Zbl 0172.05303)] \textit{R. Gilmer} proved that if \(k\subset E\) is a tower of fields such that every nonzero polynomial over \(k\) has a root in \(E\), then each polynomial in \(k[X]\) splits in \(E[X]\). In this paper, the author gives a constructive proof of a slight generalization of Gilmer's theorem.