# What does the Galois group of polynomial $f$ mean?

Michael Artin's algebra:

9.8 Porposition. let $f$ be a polynomial over $F$ whose Galois group $G$ is a simple nonabelian group. Let $F'$ be a Galois extension of $F$, with abelian Galois group. Let $K'$ be a splitting field of $f$ over $F'$. Then the Galois group $G(K'/F')$ is isomorphic to G

in my understanding , Galois is always linked with extension field. but here the Galois group is simply attached to a function definition , what does this mean ?

it is just the definition that matters.

plus can someone elaborate on the proval ?

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Okay first the answer to your question: He defines the Galois group of a polynomial $f$ over $F$ to be the Galois group of the splitting field of $f$ over $F$.