If $\Bbb K$ is an extension of $\Bbb Q$ having degree 4, why is the Galois group corresponding to the Galois closure of $\Bbb K$ a subgroup of $S_4$?
2026-04-02 23:56:43.1775174203
The Galois closure
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By the primitive element theorem, one can write $\Bbb K = \Bbb Q(\alpha)$ for some algebraic number $\alpha$ of degree $4$. The Galois group of the Galois closure of $\Bbb K$ acts on the roots of the minimal polynomial of $\alpha$; this action identifies the Galois group with a subgroup of $S_4$.