Calculating this class number

Use $f$ to find the factorization of a prime $p$ in the given ring. We have that:

$$f = x^5 \in \mathbb{F}_2[x] \implies 2\mathcal{O}_K = (2,\alpha)^5 = \mathfrak{p}_2^5$$ $$f = x^5+2x^4+1 \in \mathbb{F}_3[x] \implies 3\mathcal{O}_K = (3)$$ $$f = x^5+2x^4+3 \in \mathbb{F}_5[x] \implies 5\mathcal{O}_K = (5)$$

(The polynomials on the left are written in their factorization into irreducibles)

So from here we have that the problem reduces to proving that $\mathfrak{p}_2 = (2,\alpha)$ is a principal ideal. But this is true, as $2=\alpha(\alpha^4 + 2\alpha^3)$ and so $2 \in (\alpha)$. Hence we have that $\mathfrak{p}_2 = (\alpha)$. Hence as the class group of the ring of integers is generated by principal ideals we have that $h_K =1$