Suppose $$f_a=x^p-x-\left [\prod_{j=1}^{a-1} \alpha_j\right]^{p-1} \in \mathbb{F}_p(\alpha_1,\ldots,\alpha_{a-1})[x]$$ is irreducible over $\mathbb{F}_p(\alpha_1,\ldots,\alpha_{a-1}) $, where $\alpha_i$ is a root of $f_i$ with $i<a$. And suppose $\alpha_a$ is a root of $f_a$. How do I prove that $$\mathbb{F}_p(\alpha_1,\ldots,\alpha_a)=\mathbb{F}_p(\alpha_a)?$$
It's easy to see that $$\left [\prod_{j=1}^{a-1} \alpha_j\right]^{p-1} \in \mathbb{F}_p(\alpha_a),$$ how do we conclude that $\alpha_i \in \mathbb{F}_p(\alpha_a)$ ?