Exercise 8.2.6 (Normality)

Solution: By Theorem 7.1.5 we that $S{F}_K(f)⟺M:K$ is finite and normal, where $M=S{F}_K(f)$. For each of the extensions shown in the diagram, we can choose an $f$ e.g., $(t^2-2,t^2+1,t^2+2)$, thereby making each extension a splitting field. Therefore, we get by the theorem above, that each extension is finite and normal. Even simpler, each of the extensions is of degree 2, hence they are normal extensions.