Let $A=k[x^2,xy,y^2]\hookrightarrow B=k[x,y]$, where $k$ is a field. Is $B$ flat over $A$?
I am guessing the answer is no. My first thought is, since $B$ is integral over $A$, so it's finitely generated as an $A$-module, but I don't know how to go any further. However, I have the following theorem in hand but don't know how to apply properly in this situation.
Theorem. If $A$ is a local ring and $M$ a finitely generated flat $A$-module, then $M$ is free.