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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

REFERENCES

(1)Birkhoff, G.Lattice theory. Revised ed. (New York, 1948).Google Scholar
(2)Bourbaki, N.Topologie générale, Chap, III-IV, 2me éd. Actualités Sci. Industr. 1143 (Paris, 1951).Google Scholar
(3)Cohn, P. M. and Mahler, K.On the composition of pseudo-valuations. Nieuw Arch. Wisk. 3 (1953), 161–98.Google Scholar
(4)Dantzig, D. V.Nombres universels ou v!-adiques avec une introduction sur l'algèbre topologique. Ann. Sci. Éc. Norm. sup., Paris, (3), 53 (1936), 275–307. I have only seen the review in Zbl. Math. 16 (1937), 149–50.CrossRefGoogle Scholar
(5)Kaplansky, I.Topological methods in valuation-theory. Duke math. J. 14 (1947), 527–41.CrossRefGoogle Scholar
(6)Krull, W.Allgemeine Bewertungstheorie. J. reine angew. Math. 167 (1932), 160–92.CrossRefGoogle Scholar
(7)Lorenzen, P.Abstrakte Begründung der multiplikativen Idealtheorie. Math. Z. 45 (1939), 533–53.CrossRefGoogle Scholar
(8)Mahler, K.Über Pseudobewertungen. I. Acta math., Stockh., 66 (1935), 79–119.CrossRefGoogle Scholar
(9)Mahler, K.Über Pseudobewertungen. II. Die Pseudobewertungen eines endlichen algebraisehen Zahlkörpers. Acta Math., Stockh., 67 (1936), 51–80.CrossRefGoogle Scholar