Journal of Commutative Algebra

A theorem of Gilmer and the canonical universal splitting ring

Fred Richman

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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

Permanent link to this document
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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References

  • Michael P. Fourman and Andre Scedrov, The “world's simplest axiom of choice” fails, Manuscr. Math. 38 (1982), 325-332.
  • Robert W. Gilmer, A note on the algebraic closure of a field, Amer. Math. Month. 75 (1968), 1101-1102.
  • –––, Commutative semigroup rings, University of Chicago Press, Chicago, 1984.
  • Henri Lombardi and Claude Quitté, Algèbre commutative, Méthodes constructives, Calvage & Mounet, Paris, 2011.
  • Ray Mines, Fred Richman and Wim Ruitenburg, A course in constructive algebra, Springer, Berlin, 1988.
  • Fred Richman, van der Waerden's construction of a splitting field, Comm. Alg. 34 (2006), 2351-2356.
  • B.L. van der Waerden, Eine Bemerkung über die Unzerlegbarkeit von Polynomen, Math. Annal. 102 (1930), 738-739.
  • –––, Modern algebra, Frederick Ungar Publishing Co., New York, 1953. \noindentstyle