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 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.
 A: 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.
A: Note that both rings are integral domains of Krull dimension 5, so you have to look at slightly more subtle invariants to tell them apart.  Here is one way:
If you localize the first ring at the maximal ideal $(a,b,c,d,e,f)$, the resulting local ring is not regular.  On the other hand, all localizations of $k[x_1,\ldots,x_5]$ at its maximal ideals are regular local rings.

In geometric terms, the first ring corresponds to a cone in $6$-dim'l affine space, and in particular, it is singular at its cone point.  The second ring corresponds to $5$-dim'l affine space, which is smooth at each of its points.
