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How do you prove that $[L:K]=\deg f$ for a separable, irreducible polynomial $f \in K[X]$ whose splitting field is $L$ and for which $\text{Gal}(L/K)$ commutes ?

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Let $\alpha \in L$ be a root of $f$. It suffices to show that $K(\alpha)=L$. Suppose that $K(\alpha) \neq L$, then $\mathrm{Gal}(L/K(\alpha))$ is a subgroup of $\mathrm{Gal}(L/K)$. Since $K(\alpha)$ is not a normal extension of $K$, what can you say about it's corresponding subgroup in the Galois group? Why is this a problem?