uniqueness of an operator $p$-th root for generators of $A_n(k)$

Lemma 9   If $F\in A_n(k)$ satisfies an identity $F^p=\xi_1^p$, then $F=\xi_1$.

See the principal symbol (highest degree part) of $F$. Then it should be $\xi_1$. Thus there exists a constant $c$ such that $F=\xi_1+c$. On the other hand, we have $(\xi_1+c)^p=\xi_1^p+c^p$. $\qedsymbol$

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