Residually *-Abelian Valuation

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].

Key Words:

• Henselian valued field
• Normal quaternion
• Residually central
• Purely transcendental extension
• Residually *-abelian

2000 Mathematics Subject Classification:

• Primary 12E15
• Secondary 17A80

ACKNOWLEDGMENTS

The author wishes to express his thanks to the referee for his/her careful reading and for numerous valuable suggestions.

Notes

Communicated by M. Bresar.