TITLE: Proving that $k[a,b,c,d,e,f]/(ab+cd+ef)$ and $k[x_1,x_2,x_3,x_4,x_5]$ are not isomorphic
QUESTION [7 upvotes]: How would you show that for a field $k$, the rings $k[a,b,c,d,e,f]/(ab+cd+ef)$ and $k[x_1,x_2,x_3,x_4,x_5]$ are not isomorphic, using methods that are algebraic? 
To be quite honest, I have no idea how to approach this problem.

REPLY [5 votes]: The first variety has both singular and nonsingular $k$-points, while the second is homogeneous and nonsingular at all points.  The dimensions of the local cotangent spaces can be defined algebraically from the coordinate rings as $\dim \mathfrak{m}/\mathfrak{m^2}$ and this local invariant will be greater than $5$ at the point $(0,0,0,0,0,0)$ (ie., for $\mathfrak{m}$ the ideal generated by all the variables) for the first ring, but equal to $5$ for all maximal ideals in the second ring.