It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it.
For any $f(x_1,\dots, x_n)=\sum c_{i_1\dots i_n} x_1^{i_1}\dots x_n^{i_n}\in \mathbb{F}_{q^N}[x_1,x_2,\dots,x_n]$ with integer $N\ge 1$ and any integer $j$, we define $f^{(q^j)}(x_1,\dots, x_n)=\sum c^{q^j}_{i_1\dots i_n} x_1^{i_1}\dots x_n^{i_n}$.
Result: Let $f(x_1,x_2,\dots,x_n)$ be an irreducible polynomial of degree $M$ over $\mathbb{F}_q$. Let $N$ be an arbitrary positive integer. Assume that $f_1(x_1,x_2,\dots,x_n)$ is a monic irreducible polynomial over $\mathbb{F}_{q^N}$ satisfying that $f_1\mid f$. Then there exists $l$ dividing $M$, such that $\mathbb{F}_{q^{M/l}}$ is the smallest field containing the coefficients of $f_1$, and $$f(x_1,x_2,\dots,x_n)=c\prod_{j=0}^{M/l-1}f_1^{(q^j)}(x_1,x_2,\dots,x_n),$$ for some $c \in \mathbb{F}_q\setminus\{0\}$.