Field/Polynomial ring/Not finitely generated/Example

The polynomial ring ${}R=K[X]$ over a field ${}K$ is not a finite-dimensional vector space. To see this, we have to show that there is no finite generating system for the polynomial ring. Consider ${}n$ polynomials ${}P_{1},\ldots ,P_{n}$. Let ${}d$ be the maximum of the degrees of these polynomials. Then every ${}K$-linear combination ${}\sum _{i=1}^{n}a_{i}P_{i}$ has at most degree ${}d$. In particular, polynomials of larger degree can not be presented by ${}P_{1},\ldots ,P_{n}$, so these do not form a generating system for all polynomials.