Galois extensions for Klein four-group

Given a Galois extension of fields, the Galois group for the extension is a Klein four-group if the extension is of degree four and contains more than one quadratic extension of the base field.

Note that since Galois groups for extensions of finite fields are always cyclic, there is no example with finite fields.

Case where the characteristic of the field is not two
Where the characteristic of the field is not two, Galois extensions with Galois group the Klein four-group are extensions obtained by adjoining squareroots to two elements in the base field, such that neither of those elements is a square and their product is also not a square. (For finite fields, it is always true that one of the numbers $$a,b,ab$$ is a square, because the multiplicative group of a finite field is cyclic).

If $$K$$ is the base field, the extension field is of the form $$K(\sqrt{a},\sqrt{b})$$, and the automorphisms are:

$$\sqrt{a} \mapsto \pm \sqrt{a}, \sqrt{b} \mapsto \pm \sqrt{b}$$.

In other words, there are four automorphisms:

$$\sigma_1:\sqrt{a} \mapsto \sqrt{a}, \sqrt{b} \mapsto \sqrt{b}, \qquad \sigma_2:\sqrt{a} \mapsto -\sqrt{a}, \sqrt{b} \mapsto \sqrt{b}, \qquad \sqrt{a} \mapsto \sqrt{a}, \sqrt{b} \mapsto -\sqrt{b},\qquad \sqrt{a} \mapsto -\sqrt{a}, \sqrt{b} \mapsto -\sqrt{b}$$.

For instance, the extension $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$ is a Galois extension of degree four, and its Galois group is the Klein four-group.

The extension $$K(\sqrt{a},\sqrt{b})$$, where $$a,b,ab$$ are not squares, is also equal to the extension $$K(\sqrt{a} + \sqrt{b})$$.