Let $K $be a field and $f \in K[x]$. Then there exists a splitting field for $f$ over $K$ Let $K $ be a field and $f \in K[X]$. Then there exists a splitting field for $f$ over $K$.
I don't understand what this means, I think I am interpreting it wrongly.
Take $x^2+1 \in \Bbb{Q}[X]$ then the theorem seems to me to say that there is a splitting field for $x^2+1$ over $\Bbb{Q}$ but clearly it can't split there since the roots of $x^2+1$ are not real.
So it actually saying we can find a field extension $\Bbb{Q}\subseteq L$ such that $L$ is the splitting field? (I think $L$ here would actually be $\Bbb{Q}(i)?$)

the signification of $ L $ is the splitting field of $f$ over $K$
is: in an algebraic closure $\Omega$ of  $K$, the field $L$ is the
smallest field containing K (as minimum degree of $L$ over $K$)
and included in $\Omega$, in which $f$ split  completely (that is
has all roots in $L$)
