Multiquadratic extensions, rigid fields and pythagorean fields

Let F be a field of characteristic other than 2. Let F⁽²⁾ denote the compositum over F of all quadratic extensions of F, let F⁽³⁾ denote the compositum over F⁽²⁾ of all quadratic extensions of F⁽²⁾ that are Galois over F, and let F^{3} denote the compositum over F⁽²⁾ of all quadratic extensions of F⁽²⁾. This paper shows that F⁽³⁾ = F^{3} if and only if F is a rigid field, and that F⁽³⁾ = K⁽³⁾ for some extension K of F if and only if F is Pythagorean and K = F√-1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.