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definition- Let K be a field. The Galois group of polynomial $f \in K[x]$ is the group where F is a splitting field of $f$ over K

my problem is that galois group of polynomial unique or not

I think it is not need to be unique this is my atempt

consider $\Bbb Q \subset \Bbb Q(\sqrt2)$ and $f=x^2-2$

clearly $|gal(f)|=2 $

but when we consider $ \Bbb Q(\sqrt[4]2)$

$\Bbb Q \subset \Bbb Q(\sqrt[4]2)$ and $f=x^2-2$

$|gal(f)|\le 4 $ and $|gal(f)|= 4 $ (not sure, this equal to 4 i think there is 4 automorphims)

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Look up the definition of a splitting field again. It should say that the splitting field is the smallest field (with respect to inclusion or dimension) over which the polynomial splits, so your example will not work.
Soon after that should be a proof that the splitting field is unique up to isomorphism, giving you then that the Galois group is, too.

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  • $\begingroup$ thank you for the help $\endgroup$ – Bad English Nov 22 '17 at 11:08

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