Is Quotient Ring algebraic

Question

Let $K$ be a field and $x$ an indeterminate. Prove that the quotient ring $K[x]/(x^2)$ is algebraic. Also prove that the ring of polynomials $K[x]$ is not algebraic.

Answer 1

Hint:

The set $\{1,x,x^2,\dots\} $ is a basis for $K[x]$ as a $K$-algebra, hence $K[x]$ is infinite dimensional as a $K$- vector space. While the set $\{1+(x^2),x+(x^2)\}$ forms a basis for $K[x]/(x^2)$ as a $K$- vector space.