I've got a field extension $F_{q^n} / F_q$ of degree n and the Frobenius morphism $f$: $ x \to x^q$ associated to $F_{q^n} / F_q$. Let m be an integer dividing n.
In terms of $f$, what's an expression of the Frobenius morphism associated to $F_{q^n} / F_{q^m}$?