Cubic irreducibles over QQ are irreducible over extensions of degree 4

Let p(x) be an irreducible cubic polynomial over \mathbb{Q}. Suppose K is an extension of \mathbb{Q} of degree 4; prove that p(x) is irreducible over K.


We may assume without loss of generality that p(x) is monic.

Now suppose to the contrary that p(x) is reducible over K. Since p(x) is cubic, it must have a linear factor, and thus must have a root \alpha in K. Since \alpha is a root of the monic irreducible p(x) over \mathbb{Q}, p(x) is in fact the minimal polynomial of \alpha over \mathbb{Q}, and thus \mathsf{dim}_\mathbb{Q} \mathbb{Q}(\alpha) = 3. We thus have the following chain of fields: \mathbb{Q} \subseteq \mathbb{Q}(\alpha) \subseteq K. By Theorem 5.7 in TAN, we have that 3 divides 4, a contradiction.

So p(x) is irreducible over K.

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