Suppose $G$ is the Galois group of an irreducible degree $5$ polynomial $f \in \mathbb{Q}[x]$ such that $|G| = 10$. Then $G$ is non-abelian.
Proof: Suppose $G$ is abelian. Let $M$ be the splitting field of $f$. Let $\theta$ be a root of $f$. Consider $\mathbb{Q}(\theta) \subseteq M$. Since $G$ is abelian every subgroup is normal. This means $\mathbb{Q}(\theta) \subseteq M$ is a normal extension. So $f$ splits completely in $\mathbb{Q}(\theta)$. Then what how to complete the proof. How would I get a contradiction?