I am trying to compute the Krull dimension of $\mathbb{K}[X,Y]/ (XY+X^2+Y^3)$.
I have proved that if $R$ is a subring of $T$ and $T$ is integral over $R$, then $\dim R=\dim T$.
Hence, I do believe that I should prove that $\mathbb{K}[X,Y]/ (XY+X^2+Y^3)$ is integral over some well-known ring, such as $\mathbb{K}[X,Y]$ or $\mathbb{K}[X]$. However, I do not know how to proceed.
Can anyone give me a clue of how should I prove that $\mathbb{K}[X,Y]/ (XY+X^2+Y^3)$ is integral over some ring?
Thanks in advance.
The canonical ring morphism $$f : K[Y] \to K[X,Y] / (X^2 + YX + Y^3) = K[Y][x]$$ is integral (and injective), where $x$ denotes the class of $X$ modulo $(X^2 + YX + Y^3)$. To see why it is integral, can you give an example of a monic polynomial with coefficients in $K[Y]$ and having $x$ as a root?
Thus the Krull dimension of $K[Y][x]$ is the same as the Krull dimension of $K[Y]$, which is $1$, provided that $K$ is a field.