Abstract
Let K be a field. Then there exists a commutative K-algebra A such that each polynomial in K[X] of degree at least 2 has infinitely many roots in A. If B is a finite-dimensional commutative K-algebra and char(K) ≠ 3 (resp., char(K ) = 3), then X 2 + X + 1 (resp., X 2 + X-1) has only finitely many roots in B. Relevant examples are also given, especially of K-algebras of the form K + N, where N is the nilradical.
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Dobbs, D.E. Commutative Algebras In Which Polynomials Have Infinitely Many Roots. Results. Math. 36, 252–259 (1999). https://doi.org/10.1007/BF03322114
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DOI: https://doi.org/10.1007/BF03322114