Commutative algebras in which polynomials have infinitely many roots (Q1818225)

Given a set \(S\) of polynomials in \(K[x]\) (\(K\) a field) and for each \(f\in S\) a cardinal number \(\alpha(f)\), the author constructs a commutative algebraic \(K\)-algebra \(A\) such that every \(f\in S\) has at least \(\alpha(f)\) roots in \(A\). He also shows that there exists for every field \(K\) a quadratic polynomial (\(x^2+x+1\) if \(\chi(K)\neq 3\), \(x^2+x-1\) if \(\chi(K)=3\)) that has only finitely many roots in every finite-dimensional commutative \(K\)-algebra.