A result in deformation theory states that if every morphism Y=Spec(A)→X where A is a local Artin ring finite over k can be extended to every Y′⊃Y where Y′ is an infinitesimal thickening of Y, then X is non-singular.

My question is:

if k is algebraically closed, can we say explicitly what every local Artin ring finite over k is?

A can be of the form k[t]/(tn) or for instance if its maximal ideal isn’t principally generated k[t2,t3]/(t4). Are there A which we cannot write in this “adjoin various powers of t and mod out by some power of t” form? If there are more exotic A can we say anything non-tautological about the structure of such an A?

Thanks.

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