I was trying to read some notes leading up the Dixmier conjecture, but I was hoping to see clarify a lemma.
Suppose you have the Weyl algebra $A_n$ over a field $k$ of positive characteristic $p$. Why is the center of the Weyl algebra isomorphic to the algebra of polynomials in $2n$ variables over $k$?
A cursory remark says this follows by writing $f=\sum_{I,J}f_{IJ}x^Iy^J$ and computing $[x_i,f]$ and $[y_j,f]$ term by term.
I must be dense, but I don't see what this is getting at. Can anyone please flesh out this remark to make it more transparent why the result is true? Thank you.
Edit: The thing in question is statement (2) of Lemma 3 in these notes. I'm just hoping to see a more detailed proof than the one given.