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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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up vote 3 down vote accepted

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 $F$.

Secondly, I wanted to point out that what you refer to as Proposition 9.8 is no longer the same proposition in the new edition. In the new edition Galois Theory is Chapter 16.

Hope this helps.

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I would guess so, just to confirm. and the edition I am reading did not present Galois theory that clean. –  zinking Nov 3 '12 at 7:08
    
I did not study Galois Theory from Artin's Algebra, so I wouldn't know. You might want to read his father, Emil Artin's lucid exposition on Galois Theory. –  Rankeya Nov 3 '12 at 7:10
    
Also, no other definition of the Galois group of a polynomial makes sense, at least to me. –  Rankeya Nov 3 '12 at 7:11
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