A SORT OF POLYNOMIAL IDENTITIES OF $\[{M_n}(F)\]$ WITH CHAR $\[F \ne 0\]$

Let $F$ denote a field, finite or infinite, with characteristic $\p \ne 0\]$. In this paper, the author obtains the following result: The symmetric polynomial on $t$ letters $$\{S_{sym(t)}}({x_1},{x_2}, \cdots ,{x_t}) = \sum\limits_{x \in sym(t)} {{X_{\pi 1}}{X_{\pi 2}} \cdots {X_{\pi t}}} \]$$ is a polynomial identity of $\{M_n}(F)\]$ when $\t \ge pn\]$, and this is sharp in the sense that if $\t \le pn - 1\]$,it is not a polynomial identity of $\{M_n}(F)\]$.