Journal of Commutative Algebra

A theorem of Gilmer and the canonical universal splitting ring

Fred Richman

Abstract

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 $A[X]$.

Article information

Source
J. Commut. Algebra, Volume 6, Number 1 (2014), 101-108.

Dates
First available in Project Euclid: 2 June 2014

https://projecteuclid.org/euclid.jca/1401715580

Digital Object Identifier
doi:10.1216/JCA-2014-6-1-101

Mathematical Reviews number (MathSciNet)
MR3215563

Zentralblatt MATH identifier
1291.12001

Citation

Richman, Fred. A theorem of Gilmer and the canonical universal splitting ring. J. Commut. Algebra 6 (2014), no. 1, 101--108. doi:10.1216/JCA-2014-6-1-101. https://projecteuclid.org/euclid.jca/1401715580

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