= Quadratic subfields of a multiquadratic extension
{title2=$L=K(\prod_{i\in I}x_i)$}
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.
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