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So I got the Witt Algebra over finite field http://en.wikipedia.org/wiki/Witt_algebra and need to show that it's simple.

But, I don't know where to start. So let I be a nonzero ideal of witt algebra, but then don't know where to go with this.

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    Maybe if you give some more idea of how much you know about this algebra it would help. Do you know how elements multiply, for example?2012-04-18

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For characteristic at least 3, there is a proof in the book "Modular Lie Algebras and their Representations" by Strade and Farnsteiner.

For characteristic 2, in the cases where it actually is simple, I have a proof written up somewhere, but don't have it on this computer. The general idea is to take an arbitrary element of your ideal and then show that every element of the Witt algebra can be expressed as a product of that element with some other element.

EDIT: It turns out what I was thinking of was the generalised Jacobson-Witt algebra, of which the Witt algebra is a special case, but I can't remember which one. So probably your question is a lot easier than I was thinking.