The splitting field of a polynomial $p$ with coefficients over a field $K$ is defined as the smallest field that contains $K$, in which the polynomial can split into linear factors.
I just want to see if I've understood the definition correctly. Does the splitting field depend not only on the polynomial but also on the field $K$ that we choose? In the sense that, the splitting field of $p=t^2+1$ over $\mathbb{Q}$ is $\mathbb{Q}(i)$, but the splitting field of the same polynomial over $\mathbb{R}$ would be $\mathbb{R}(i)$?
$\endgroup$ 31 Answer
$\begingroup$(commenting so you can mark as closed)
Yes this is (except for the typo mentioned in the comments) completely correct.
$\endgroup$
