Commutative Algebras In Which Polynomials Have Infinitely Many Roots

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.