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Suppose $K/F$ is a degree $n$ separable extension of fields, and $G\leq S_n$ is the Galois group of its Galois closure $K^{gal}$. For each field $F\subseteqq E \subsetneqq K$ which is Galois over $F$, take a minimal degree field $E'$ such that $F\subseteqq E'\subseteqq E$ and $E$ is the Galois closure of $E'/F$. Then $\Gal(E/F)$ has a natural embedding as a Galois group $H\leq S_m$ with $m=[E':F]$. These Galois groups $H$ are resolvents of $G$.

In group theoretic terms, the groups $H$ correspond to subgroups $A\leq G$ such that the core of $A$ is non-trivial. Then $G$ acts transitively on the left cosets of $A$ by left translation, giving a homomorphism $\rho:G\to S_m$ where $m=[G:A]$. Then $H=\text{Im}(\rho)$.

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  • Last edited by Kiran S. Kedlaya on 2019-05-03 16:37:43
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