Geodesic
On the Number of Prolongations of a Finite Rank Valuation
Canadian journal of mathematics, Tome 23 (1971) no. 3, pp. 553-556

Voir la notice de l'article provenant de la source Cambridge University Press

A (non-archimedean) valuation v on a field K is said to be henselian if it has a unique prolongation to a valuation on Ka, the algebraic closure of K. A henselization (Kh, vh) of a valuated field (K, v) is a smallest separable extension of K containing a henselian prolongation vh of v. (Kh, vh) is unique up to K-isomorphism, and (Kh, vh) = (K, v) if and only if v is henselian. In this paper we confine ourselves to valuations of finite rank.If v is a non-henselian rank one valuation on K, and if [Ks:K] = ∞, Ks being the separable closure of K, then it is known that v has infinitely many prolongations to Ka [1, (27.11)]. We shall see that this is no longer true if the rank of v is greater than one. 
Wright, Michael J. On the Number of Prolongations of a Finite Rank Valuation. Canadian journal of mathematics, Tome 23 (1971) no. 3, pp. 553-556. doi: 10.4153/CJM-1971-061-0
@article{10_4153_CJM_1971_061_0,
     author = {Wright, Michael J.},
     title = {On the {Number} of {Prolongations} of a {Finite} {Rank} {Valuation}},
     journal = {Canadian journal of mathematics},
     pages = {553--556},
     year = {1971},
     volume = {23},
     number = {3},
     doi = {10.4153/CJM-1971-061-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1971-061-0/}
}
TY  - JOUR
AU  - Wright, Michael J.
TI  - On the Number of Prolongations of a Finite Rank Valuation
JO  - Canadian journal of mathematics
PY  - 1971
SP  - 553
EP  - 556
VL  - 23
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1971-061-0/
DO  - 10.4153/CJM-1971-061-0
ID  - 10_4153_CJM_1971_061_0
ER  - 
%0 Journal Article
%A Wright, Michael J.
%T On the Number of Prolongations of a Finite Rank Valuation
%J Canadian journal of mathematics
%D 1971
%P 553-556
%V 23
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1971-061-0/
%R 10.4153/CJM-1971-061-0
%F 10_4153_CJM_1971_061_0

[1] 1. Endler, O., Bewertungstheorie. Unter Benutzung einer Vorlesung von W. Krull, Vols. I and II, Bonn. Math. Schr. No. 15 (1963). Google Scholar

[2] 2. Endler, O., A note on henselian valuation rings, Can. Math. Bull. 11 (1968), 185–189. Google Scholar

[3] 3. Kaplansky, I. and Schilling, O. F. G., Some remarks on relatively complete fields, Bull. Amer. Math. Soc. 48 (1942), 744–747. Google Scholar

[4] 4. Ribenboim, P., A short note on henselian fields, Math. Ann. 178 (1967), 83–88. Google Scholar

[5] 5. Ribenboim, P., Théorie des valuations (Sem. Math. Sup., Université de Montréal, 1964). Google Scholar

[6] 6. Rim, D., Relatively complete fields, Duke J. Math. 24 (1957), 197–200. Google Scholar

Cité par Sources :