An interpretation of the corollary might be:

If $S$ is over a number field $F$, $B$ can be over a number field $E$, and $F$ need not be the same as $E$.

Specifically, If $S$ is real valued, $B$ can be complex valued. Hence $\sqrt{D_kD_{k-1}}$ will be always defined; i.e. $D_kD_{k-1}$ need not be positive.