# Why is the Weyl Algebra simple?

Let $\mathbb K$ be a field and $L=\mathbb K\langle x, \frac{d}{dx}\rangle$ be the weyl algebra over the field $\mathbb K$. That is, the algebra over $\mathbb K$ with two generators $x$ and $\frac{d}{dx}$ and one relation $\left[x , \frac{d}{dx}\right]=1$. Where can I find a proof that this algebra is simple?

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This is proved in pretty much any textbook which introduces the Weyl algebra. Where have you looked? –  Mariano Suárez-Alvarez Nov 23 '12 at 22:23
(By the way, the algebra is simple if and only if the field $\mathbb k$ is of characteristic zero) –  Mariano Suárez-Alvarez Nov 23 '12 at 22:24
Gracias Mariano. Es que algebras de Weyl es algo muevo para mi y los libros que he mirado no tenian esta demonstración. –  zacarias Nov 23 '12 at 22:28
Also, the relation is $[x,\tfrac{d}{dx}]=1$, not what you wrote. –  Mariano Suárez-Alvarez Nov 23 '12 at 22:38
@MarianoSuárez-Alvarez: Che flaco: what is a good book to study properties of the Weyl algebra that has a more applied (computational algebraic geometry) focus? Thanks for any help. –  Sergio Parreiras Dec 9 '13 at 16:32