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# Galois group

The *Galois group* $\operatorname{Gal}(K/F)$ of a field extension $K/F$ is the group of all field automorphisms $\sigma\colon K\to K$ of $K$ which fix $F$ (i.e., $\sigma(x)=x$ for all $x\in F$). The group operation is given by composition: for two automorphisms $\sigma_{1},\sigma_{2}\in\operatorname{Gal}(K/F)$, given by $\sigma_{1}\colon K\to K$ and $\sigma_{2}\colon K\to K$, the product $\sigma_{1}\cdot\sigma_{2}\in\operatorname{Gal}(K/F)$ is the composite of the two maps $\sigma_{1}\circ\sigma_{2}\colon K\to K$.

The *Galois group* of a polynomial $f(x)\in F[x]$ is defined to be the Galois group of the splitting field of $f(x)$ over $F$.

Related:

FundamentalTheoremOfGaloisTheory, InfiniteGaloisTheory

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Definition

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## Mathematics Subject Classification

12F10*no label found*

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