Abstract
We are given a division ring D with involution (*) and with a *-valuation V such that V(aa*b − baa*) >0 V(aa*b), for all nonzero elements a, b of D. We assume, further, that the associated residue class division ring D V is a (commutative) field with characteristic 0. In this work, we evidence two criteria in order for D to be either a standard quaternion division algebra or else a purely transcendental extension of its center. We apply one of these to answer our open Question 2.4.3 [2].
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