# Is Galois group of polynomial unique?

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)