A field extension is Galois, if it’s normal and separable.
Every field extension of degree 2 is normal. Proof
We need to find a field extension of degree 2 that is not separable.
\(\mathbb{Z}_2 (x) : \mathbb{Z}_2 (x^2)\) is not separable, because the minimal polynomial $$y^2+x^2$$ of $$x$$ over $$\mathbb{Z}_2 (x^2)$$ has in $$\mathbb{Z}_2 (x)$$ the double zero $$x$$ because $$y^2+x^2=(y+x)^2$$