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What can be said about the structure of a finite dimensional, commutative, associative, unital local algebra over an algebraically closed field of characteristic zero?

Frank
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    Related question: http://math.stackexchange.com/questions/384902/classification-of-local-artin-commutative-rings-which-are-finite-over-an-algeb – 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
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    You 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/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.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.