# zbMATH — the first resource for mathematics

The algebraic closure of a $$p$$-adic number field is a complete topological field. (English) Zbl 1141.12002
Summary: The algebraic closure of a $$p$$-adic field is not a complete field with the $$p$$-adic topology. We define another field topology on this algebraic closure so that it is a complete field. This new topology is finer than the $$p$$-adic topology and is not provided by any absolute value. Our topological field is a complete, not locally bounded and not first countable field extension of the $$p$$-adic number field, which answers a question of Mutylin.
##### MSC:
 12J99 Topological fields
##### Keywords:
topological field; $$p$$-adic field
Full Text:
##### References:
 [1] ALEXANDRU V.-POPESCU N.-ZAHARESCU A.: On the closed subfields of $$C^p$$. J. Number Theory 68 (1998), 131-150. · Zbl 0901.11035 [2] AMICE Y.: Les nombres p-adiques. Collection SUP. Le mathematicien 14, Presses Universitaires de France, Paris, 1975. · Zbl 0313.12104 [3] ARNAUTOV V. I.-GLAVATSKY S. T.-MIKHALEV A. V.: Introduction to the Theory of Topological Rings and Modules. Marcel Dekker, New York, 1996. · Zbl 0842.16001 [4] GOUVEA F. Q.: p-Adic Numbers: An Introduction (2nd. Springer-Verlag, New York, 1997. · Zbl 0874.11002 [5] IOVITA A.-ZAHARESCU A.: Completions of r. a.t.-valued fields of rational functions. J. Number Theory 50 (1995), 202-205. · Zbl 0813.12006 [6] KOBLITZ N.: p-Adic Numbers, p-Adic Analysis, and Zeta-Functions (2nd. Springer-Verlag, New York, 1984. · Zbl 0364.12015 [7] LAMPERT D.: Algebraic p-adic expansions. J. Number Theory 23 (1986), 279-284. · Zbl 0586.12021 [8] MARCOS J. E.: Lacunar ring topologies and maximum ring topologies with a prescribed convergent sequence. J. Pure Appl. Algebra 162 (2001), 53-85. · Zbl 1094.13542 [9] MARCOS J. E.: Locally unbounded topological fields with topological nilpoients. Fund. Math. 173 (2002), 21-32. · Zbl 1052.12005 [10] MARCOS J. E.: Erratum to Locally unbounded topological fields with topological nilpotents. Fund. Math. 176 (2003), 95-96. [11] MURTY M. R.: Introduction to p-Adic Analityc Number Theory. Amer. Math. Soc, Providence, RI, 2002. [12] MUTYLIN A. F.: Connected, complete, locally bounded fields. Complete not locally bounded fields. Math. USSR Sbornik 5 (1968), 433-449 [ · Zbl 0181.32701 [13] POONEN B.: Maximally complete fields. Enseign. Math. 39 (1993), 87-106. · Zbl 0807.12006 [14] ROBERT A.M.: A Course in p-Adic Analysis. Springer-Verlag, New York, 2000. · Zbl 0947.11035 [15] SHELL N.: Topological Fields and Near Valuations. Marcel Dekker, New York, 1990. · Zbl 0702.12003 [16] SCHIKHOF W. H.: Ultrametric Calculus. An introduction to p-adic analysis. Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge, 1984. · Zbl 0553.26006 [17] WIESLAW W.: Topological Fields. Marcel Dekker, New York, 1988. · Zbl 0661.12011 [18] ZELENYUK E. G.-PROTASOV I. V.: Topologies on abelian groups. Math. USSR Izvestiya 37 (1991), 445-460. · Zbl 0728.22003 [19] ZELENYUK E. G.-PROTASOV I. V.-KHROMULYAK O. M.: Topologies on countable groups and rings. Dokl. Akad. Nauk Ukrain. SSR 182 no. 8 (1991), 8-11.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.