Valuations on Algebraic Division Algebras

ABSTRACT

Let D be a division algebra algebraic over its center F. Given a (Krull) valuation v on F, it is shown that v extends to a valuation on D if and only if for each separable element c∈D^′ there exists a valuation w on K: = F(c) extending v on F such that $K\cap D^{\prime }\subset W^{\ast }$, where D^′ is the derived group of D* and W* is the unit group of the valuation ring W of w.

KEYWORDS:

• Division ring
• field
• uniqueness
• valuation

MATHEMATICS SUBJECT CLASSIFICATION:

• 16K20
• 16K40
• 12B10
• 12J20

Acknowledgements

The authors would like to thank the referee for his/her valuable suggestions as well as A. R. Wadsworth for his constructive comments on an earlier version of this note. The first author thanks the Research Council of the University of Garmsar and the second author thanks the one of Sharif University of Technology for support.