The Ordinary Quaternions Over a Pythagorean Field

Abstract

Let L be a proper finite Galois extension of a field K and let D be a division algebra with center K. If every subfield of D properly containing K contains a K-isomorphic copy of L, it is shown that K must be Pythagorean, $L \cong K(\sqrt { - 1} )$, and D is the ordinary quaternions over K. If one assumes only that every maximal subfield of D contains a K isomorphic copy of L, then, under the assumption that [D : K] is finite, it is shown that K is Pythagorean, $L = K(\sqrt { - 1} )$, and D contains the ordinary quaternions over K.