Quaternion fields inside Pythagorean closure
In J. Pure Appl. Algebra 30, 95-107 (1983; Zbl 0518.10024)], \textit{R. Ware} proved the following theorem: If F is a formally real field and \(G=G_ F(py)\) is the Pythagorian Galois group of F (i.e. the Galois group of the smallest algebraic extension, F(py), of F in which every sum of squares is a square). Then the following properties are equivalent: (1) Every extension of F, of degree 4, contained in F(py), is normal over F. (2) The dihedral group of order 8 is not a homomorphic image of G. In the paper the author shows that actually (1) and (2) are also equivalent to the statement: Every homomorphic image of degree \( 8\) of the group G is abelian.
scientific article; zbMATH DE number 4110816