The **Galois group** of an irreducible separable polynomial of degree $n$ can be embedded in $S_n$ through its action on the roots of the polynomial, with the image being well-defined up to labeling of the roots. Different labelings lead to conjugate subgroups. The subgroup acts transitively on $\{1,\ldots,n\}$. Conversely, for every transitive subgroup $G$ of $S_n$ with $n\in\mathbb{Z}^+$, there is a field $K$ such that $G$ is the Galois group of some polynomial over $K$.

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- Last edited by Kiran S. Kedlaya on 2019-05-02 23:34:24

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