What can be said about the structure of a finite dimensional, commutative, associative, unital local algebra over an algebraically closed field of characteristic zero?
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3Related question: http://math.stackexchange.com/questions/384902/classificationoflocalartincommutativeringswhicharefiniteoveranalgeb – Julian Rosen Jul 21 '13 at 19:58

That question is only about $0$dimensional rings and it is still an open problem. This one is a bit harder. – Matt Jul 21 '13 at 21:34

2You might want this: http://en.wikipedia.org/wiki/Cohen_structure_theorem – Marci Jul 21 '13 at 21:55
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This isn't a full answer by any means, but I might suggest reading up through Theorem 2.1 in this paper
http://www.ams.org/journals/tran/201236409/S000299472012054304/S000299472012054304.pdf
Let's call our field $k$. The upshot is that describing all of the $k$algebras with the properties you list is equivalent, in some sense, to describing all of the submodules with finite $k$dimension of a certain module over a power series ring.
Theorem 2.1 refines this notion a bit; it gives us an equivalent formulation of the problem of describing all of these $k$algebras up to $k$algebra isomorphism.