I am trying to prove the following statement stated in the proof of another theorem:
Given the finite field extensions $\mathbb{F}_p\subset \mathbb{F}_{p^d} \subset \mathbb{F}_{p^n}$, there are $d$ ring homomorphisms $\mathbb{F}_{p^d}\to \mathbb{F}_{p^n}$.
The closest result I could find on StackExchange is the following:
Let $\mathbb{Q}$ $\subset$ $K$ $\subset$ $\mathbb{C}$ with $[K:\mathbb{Q}]$ finite. Then the number of field homomorphisms from $K$ to $\mathbb{C}$ is equal to $[K: \mathbb{Q}]$. [Proof]
My attempt:
Since the exensions over prime subfield are simple, $\mathbb{F}_{p^d} = \mathbb{F}_p(\alpha)$ and $\mathbb{F}_{p^n}=\mathbb{F}_p(\beta)$ for some $\alpha$ and $\beta$ in $\overline{\mathbb{F}_p}$. Then the $d$ homomorphisms are given by identity map and $\alpha^j\mapsto \beta$ for $1\leq j\leq d-1$. Existence of any other map should contradict the injective property of homomorphism, but how? I am not sure whether my argument makes sense.