Polynomial quotient ring : k[x,y,z,t]/(xy-zt) I have some trouble picturing a quotient. Namely, what $k[x,y,z,t]/(xy-zt)$ looks like where $k$ is a field ?
My intuition is probably wrong but is it isomorphic to $k[u,v,w]$ ?
 A: The ring $A=k[x,y,z,t]/(xy-zt)$ is the ring of functions of the affine quadratic cone $V\subset \mathbb A^4$ with equation $xy-zt=0$.     
The dictionary at the heart of algebraic geometry  immediately shows you that the ring $A$ is not isomorphic to $k[u,v,w]$ because that cone  $V$ is not smooth at the origin and thus cannot be isomorphic to $\mathbb A^3$, the variety associated by the dictionary to $k[u,v,w]$, which is smooth everywhere.
Non smoothness  of $V$ at the origin is shown by the Jacobian criterion (the same  as in advanced calculus) and boils down to the fact that the gradient $(y,x,-t,-z)$  of $xy-zt$ vanishes at the origin $O=(0,0,0,0)$.
[Actually all cones are singular at the origin, except if they are linear subspaces, and this allows you to show that many similar quotient rings  are not isomorphic to polynomial rings]   
Of course there are also purely algebraic methods for proving that $A$ is not isomorphic to a ring of polynomials: Martin indicated  one in his comment.
But I hope that you appreciate how elegantly algebraic geometry can address such questions.
