Here I am taking the definition of isomorphism to be an injective homomorphism.
Suppose we have two finite fields of the same characteristic, $\mathbb{F}_{p^n}$ and $\mathbb{F}_{p^m}$ with $m<n$. Is there always an isomorphism $\phi:\mathbb{F}_{p^m}\rightarrow \mathbb{F}_{p^n}$?
If $m|n$ then we could use the same construction $\mathbb{F}_{p^m}[X]/(P)$ where $P$ is an irreducible polynomial in $\mathbb{F}_{p^m}[X]$ such that $\deg(P)m=n$. Since this would produce a field of the same characteristic and cardinality, it would provide an isomorphism.
Does anyone have any references on this topic? Thanks in advance.
There is a homomorphism (note: all homomorphisms of fields are injective) $\mathbb{F}_{p^n} \to \mathbb{F}_{p^m}$ exactly when $n\mid m$.
You have already shown one direction. For the other, assume that such a map exists. Then $\mathbb{F}_{p^m}$ is naturally a vector space over $\mathbb{F}_{p^n}$, which implies that $|\mathbb{F}_{p^m}| = |\mathbb{F}_{p^n}|^k$ for some $k\geq 1$, so that $p^m = p^{kn}$, therefore $m=kn$.