(D&F Example (2) at p.563) More generally, any quadratic extension $K$ of any field $F$ of characteristic different from $2$ is Galois.
(D&F Example (4) at p.563) The extension $\mathbb{Q}(\sqrt{ 2 }, \sqrt{ 3 })$ is Galois over $\mathbb{Q}$ because it is the splitting field of $(x^{2} - 2)(x^{2} - 3)$.
(D&F Example (6) at p.566) The field $\mathbb{Q}(\sqrt[4]{ 2 })$ is not Galois over $\mathbb{Q}$ while we have $\mathbb{Q} \subset \mathbb{Q}(\sqrt{ 2 }) \subset \mathbb{Q}(\sqrt[4]{ 2 })$ and both $\mathbb{Q}(\sqrt{ 2 }) / \mathbb{Q}$ and $\mathbb{Q}(\sqrt[4]{ 2 }) / \mathbb{Q}(\sqrt{ 2 })$ are Galois.