Please, I need some clarification about Weyl algebra, in Lemma 3, statment 1 and 2 Weyl algebra:
For any ring $k$, we denote by $A_n(k)$ the Weyl algebra. $$ A_n(k)=k\left\langle\xi_1, \xi_2, \ldots, x_n, \eta_1, \eta_2, \ldots, \eta_n\right\rangle /\left(\eta_j \xi_i-\xi_i \eta_j-\delta_{i j} ; 1 \leq i, j \leq n\right) $$
Lemma 3. Let $k$ be a field of characteristic $p$. We have the following facts.
(1) $\xi_i^p, \eta_j^p$ belongs to the center of $A_n(k)$.
(2) More precisely, the center $Z_n(k)=Z\left(A_n(k)\right)$ of the ring $A_n(k)$ is given by $$ Z_n(k)=k\left[\xi_1^p, \ldots, \xi_n^p, \eta_1^p, \ldots, \eta_n^p\right] $$
To prove this is the centre of weyl algebra.
My attempt is: let $n=1$, then: $A_{1}(k)=k [\xi, \eta]$, So the centre of $A_{1}$ according to 2 is $$ Z(A_1(k))=K \left[\xi^p, \eta^p\right] $$ to prove this we suppose: $\xi^p$ is central, then: $\left[\xi, \xi^p\right]=p \xi_1^{p-1}=0$
My question: after this step, please, how I can say this element is central? I mean why $p \xi_1^{p-1}=0$
I would appreciate if you could help.