Example. $\mathbb{Q}(\sqrt[3]{ 2 }) / \mathbb{Q}$ and $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}$ [2209]

    1. $\mathbb{Q}(\sqrt[3]{ 2 }) / \mathbb{Q}$ is not normal. $m_{\sqrt[3]{ 2 }, \mathbb{Q}}(x) = x^{3} - 2$ does not split completely in $\mathbb{Q}(\sqrt[3]{ 2 })$ since the other two roots $\sqrt[3]{ 2 } \frac{-1 \pm \sqrt{ -3 }}{2}$ do not belong to $\mathbb{Q}(\sqrt[3]{ 2 })$.
    2. $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}$ is normal.
      1. Use (3) of Proposition 2205: $m_{\sqrt[3]{ 2 }, \mathbb{Q}}(x) = x^{3} - 2$ and $m_{\sqrt{ -3 }, \mathbb{Q}}(x) = x^{2} + 3$ both split completely over $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 }) / \mathbb{Q}$.
      2. Use (4) of Proposition 2205: $\mathbb{Q}(\sqrt[3]{ 2 }, \sqrt{ -3 })$ is the splitting field of $x^{3} - 2 \in \mathbb{Q}[x]$.