Let $\alpha \in \mathbb{C}$ be an algebraic element on $\mathbb{Q}$ with minimal polynomial $p_\alpha$ and let $\beta \in \mathbb{C}$ be another root of $p_\alpha$. For $f \in \mathbb{Q}[x]$ let $$\begin{split} \sigma : \mathbb{Q}(\alpha) &\rightarrow \mathbb{C}\\ f(\alpha) &\mapsto f(\beta). \end{split}$$
I want to know if $\sigma$ is a field homomorphism which is identity on $\mathbb{Q}$. However I don't know where to start because we only have one mapping specified in the definition.
The two evaluation maps $$\varepsilon_{\alpha}:\ \Bbb{Q}[X]\ \longrightarrow\ \Bbb{C}:\ X\ \longmapsto\ \alpha,$$ $$\varepsilon_{\beta}:\ \Bbb{Q}[X]\ \longrightarrow\ \Bbb{C}:\ X\ \longmapsto\ \beta,$$ are injective ring homomorphisms, both with kernel $p_a$. Their images in $\Bbb{C}$ are the subrings $\Bbb{Q}(\alpha)$ and $\Bbb{Q}(\beta)$, and so it follows that $$\Bbb{Q}(\alpha)\cong\Bbb{Q}[X]/(p_a)\cong\Bbb{Q}(\beta).$$ This composition of isomorphisms maps $\alpha$ to $\beta$, so it is your map $\sigma$.