Show that $A \cong \mathbb{C}^n$ with A a commutative algebra 
Let A be a commutative algebra of finite dimension, and if $A$ has no nilpotent elements other than $0$, is true that $A \cong  \mathbb{C}^n$ ? 

The question emerge to my mind, I thought that the finite dimension tell us that the scheme is Artinian (geometrically dimension 0).
I think the pattern is just a meeting of $n$ points but I have not managed to prove it.
Someone can enlighten me please ?
Thanks

A finitely generated commutative algebra with no nilpotents is always the ring of regular functions on some affine variety.  Since your algebra is finite dimensional it is artinian and therefore has only finitely many maximal ideals, so since by the nullstellensatz maximal ideals are in 1-1 correspondence with points, it must be the ring of regular functions on some finite point space.  The ring of regular functions on an $n$ point space is just $\mathbb{C}^n$
