TITLE: Why is every quadratic subfield of a Galois extension of the rationals with the quaternions as Galois group real? 
QUESTION [6 upvotes]: Suppose that L is a field extension of the rationals with Galois group the quaternions Q={1,-1,i,-i,j,-j,k,-k}. Furthermore assume that L contains a quadratic subfield K. I have learned from this link 
http://www.math.princeton.edu/generals/bhargava_manjul 
that K is real. How can we prove this result? Of course one should use Galois theory There are three subgroups inside Q of order four. Each of them is cyclic and generated by i, j, or k, resp. But I don't know how to move on. Any hint is appreciated.

REPLY [2 votes]: A theorem of Witt asserts that a biquadratic extension $F(\sqrt{a},\sqrt{b}),\; a,b\in F^{\times}$ can be embedded in a quaternion Galois extension of $F$ iff the quadratic form $ax^2+by^2+abz^2$ is isomorphic over $F$ to $x^2+y^2+z^2$.  For $F=\mathbb{Q}$ this forces $a,b,ab>0$.