CENTRALIZERS IN THE FIRST WEYL ALGEBRA OVER A 2 OR 3-CHARACTERISTIC FIELD

The purpose of this paper is the determination of some centralizers in $A_{1}$, the first Weyl Algebra. Some authors have done their studies in the case of zero characteristic field. As far as we're concerned, we have decided to work in 2 or 3 characteristic field. Doing so, we show that if $u\in A_{1}$ is a minimal element, $C$-primitive and without constant term, then its centralizer $Z(u)=\mathbb{L}[u]\cap A_{1}$ where $\mathbb{L}$ is the fractions field of $C$, the center of $A_{1}$. Particularly, when $u$ is ad-invertible, i.e there exists $v\in A_{1}$ such that $[u,v]=1$, then we have $Z(u)=C[u]$ which is a result analogous to that of \cite{JJC}.