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An invariant characterization of pseudo-valuations on a field

Published online by Cambridge University Press:  24 October 2008

P. M. Cohn
P. M. Cohn
Affiliation:
The UniversityManchester 13
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Extract

This note is concerned with pseudo-valuations on a (commutative) field F. A pseudo-valuation on F† is defined as a real-valued function W on F such that

(1) W(x) ≥ 0 for all x∈F and W(0) = 0, but W does not vanish identically on F,

(2) W(xy) ≤ W(x) W(y) for all x, y ∈ F,

(3) W(x−y) ≤ W(x) + W(y) for all x, y ∈ F.

W is non-Archimedean, if it satisfies the ultrametric inequality

(3′) W(x−y) ≤ max {W(x), W(y)}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 159 - 177
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

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