Multiquadratic field ($K(\sqrt{a_1},\dots ,\sqrt{a_r})$)

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 $K$.

In characteristic different from two, a field generated by finitely many nonzero square roots is the splitting field of a collection of separable quadratic polynomials. Its Galois group embeds into the sign-vector group $(\mathbb{Z}/2{)}^n$. Every index-two subgroup is the kernel of a nontrivial sign character. Coordinate sign characters span the dual of this subgroup of $(\mathbb{Z}/2{)}^n$, by extending a linear functional to the ambient vector space. A product of the corresponding roots transforms by that character and generates precisely the quadratic fixed field.