 Multiquadratic field

A field extension obtained by adjoining finitely many square roots in characteristic different from two. If the classes of $a_1,\dots ,a_r$ are independent in $K^{\times }/(K^{\times }{)}^2$, the extension has degree $2^r$ and Galois group $(\mathbb{Z}/2\mathbb{Z}{)}^r$: each square root's sign can be changed independently. The products of distinct square roots form a basis over $K$.