Let $GL_n(\mathbb F_q)$ be the General linear group over a finite field $\mathbb F_q$, where $q$ is any prime power. Then how can I show that for $1 \leq m \leq n$ there is an element of order $q^m-1$ in $GL_n(\mathbb F_q).$
I know that the maximal possible order of an element in $GL_n(\mathbb F_q)$ is $q^n-1,$ which follows from the fact that order of an invertible matrix in $GL_n(\mathbb F_q)$ is same as the order of the minimal polynomial, and order of an irreducible polynomial $f(X) \in \mathbb F_q[X]$ with $f(0) \neq 0$ and $\text {deg}(f)=m$ is same as the order of any (hence all) root of $f$ in $\mathbb F_{q^m}^{\times}.$
In this problem since $\mathbb F_{q^m}$ is not necessarily a subfield of $\mathbb F_{q^n}$ for $1 \leq m \leq n,$ I cannot prove it. I need some help. Thanks.
