Let $k=\mathbb{C}(t)$ be the field of rational functions over $\mathbb{C}$.
I want to show that $P(x)=x^2+t \in k[x]$ is irreducible over $k$, and further find the degree of the splitting field of $P(x)$ over $k$.
I know that there is not a 'one size fits all' method to show irreducibility and that there are many different irreducibility criterion, but I am not sure what to do here.
Are there any suggestions for what to do here? Thanks!