Find the degree of the splitting field for the polynomial x^4 + 19 over the field Q(root19).
$\displaystyle x^4+19=(x^2+\sqrt{19}\,i)(x^2-\sqrt{19}\,i)$ ; since the field $\displaystyle \mathbb{Q}(\sqrt{19})$ is a real field, the eement $\displaystyle i=\sqrt{-1}$ is not contained in it...try to take it from here.