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If $P$ and $Q$ are commuting elements of the first Weyl algebra, over some field $k$, is it true that there exists an element $H$ in the Weyl algebra such that $P$ and $Q$ are polynomials in $H$ with coefficients from $k$?

I am nearly certain that this is not true but I have not found a counterexample.

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up vote 2 down vote accepted

Consider the first Weyl algebra $A=k\langle x,y \rangle/(yx-xy-1)$ over $k$ a field of characteristic $p$. Then $x^p$ and $y^p$ are central, so for example $x^py=yx^p$, but $x^p$ and $y$ do not satisfy the conditions as given.

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Good example, but I have realized that I would really like to know the answer when the ground field has characteristic zero. Sorry about that. I will accept your answer and ask a new question. –  Johan Oct 10 '12 at 11:17
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