There is no $n$ such that every ideals of $K[X,Y]$ is generated by $n$ elements

Question

How to prove that there does not exist any integer $n$ such that all ideals of $K[X,Y]$ can be generated by $n$ elements?

Answer 1

Look at the ideal of all polynomials that have no terms of degree $n-1$ or less. This contains the $n+1$ monomials $x^n,x^{n-1}y,\ldots,xy^{n-1},y^n$ (it is actually generated by them but we will not use this fact).

If there existed a set $\{g_1,\ldots,g_n\}$ of generators then each of these monomials would be a linear combination of the $n$ generators with coefficients in $K[X,Y]:$

$$x^jy^{n-j}=\sum_{i=1}^na_{ij}(x,y)g_i(x,y)$$

The constant terms of the polynomials $a_{ij}$ define a linear transformation from the $K$-linear span of the $n$-th degree parts of the $g_i$ (which has dimension at most $n$) onto the $K$-linear span of the $n+1$ linearly independent $x^jy^{n-j}$, which is impossible.