×

Completion of r. t. extensions of local fields. II. (English) Zbl 0923.12010

Let \((K,v)\) be a local field. An extension \(w\) of \(v\) to \(K(X)\) is called residual transcendental (r.t.-extension) if the residue field of \(w\) is a transcendental extension of the residual field of \(v\). In the first part of this paper [Math. Z. 221, 675-682 (1996; Zbl 0852.12003)] the authors described the completion \((\widetilde{K(X)},\widetilde w)\) of \((K(X),w) \), where \(w\) is the Gauss r.t.-extension of \(v\). In this paper they consider the general case of any r.t.-extension of \(v\). For the description of elements of the completion they use the embedding of \((K(X),w)\) into the valued field \(K' \{\{X-a.\delta\}\}\) of \(\delta\)-formal Laurent series over the local field \((K', v')\), where \(\delta\) is an appropriate rational number, and \(K'=K(a) \subset \overline K\), \(a\in\overline K\) (the algebraic closure of \(K)\). Moreover, they describe \((\widetilde{K(X)},\widetilde w)\) as a finite extension of \((\widetilde {K(r)}, \widetilde{w_0})\), where \(r\in K(X)\) and \(w_0\) is the Gauss r.t.-extension of \(v\).

MSC:

12J10 Valued fields
12F99 Field extensions
12J20 General valuation theory for fields

Citations:

Zbl 0852.12003
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] V. Alexandru - A. POPESCU - N. POPESCU, Completion of r.t. extensions of local fields (I) , Math. Z. , 221 ( 1996 ), pp. 675 - 682 . Article | MR 1385174 | Zbl 0852.12003 · Zbl 0852.12003 · doi:10.1007/PL00004526
[2] V. Alexandru - N. POPESCU, Some elementary remarks about n-local fields , Rend. Sem. Mat. Univ. Padova , 92 ( 1994 ), pp. 17 - 28 . Numdam | MR 1320475 | Zbl 0846.12006 · Zbl 0846.12006
[3] V. Alexandru - N. POPESCU - A. ZAHARESCU, A theorem of characterization of residual transcendental extensions of a valuation , J. Math. Kyoto Univ. , 28 , No. 4 ( 1988 ), pp. 579 - 592 . Article | MR 981094 | Zbl 0689.12017 · Zbl 0689.12017
[4] V. Alexandru - N. POPESCU - A. ZAHARESCU, Minimal pairs of definition of a residual transcendental extension of a valuation , J. Math. Kyoto Univ. , 30 , No. 2 ( 1990 ), pp. 207 - 225 . Article | MR 1068787 | Zbl 0728.12009 · Zbl 0728.12009
[5] N. Boursaki , Algébre Commutative, Ch. V, VI , Herman , Paris ( 1964 ).
[6] A.N. Parshin , Abelian covering of arithmetic schemes (russian) , Dokl. Acad. Nauk , SSSR , 273 , No. 4 ( 1978 ), pp. 855 - 858 . MR 514485 | Zbl 0443.12006 · Zbl 0443.12006
[7] L. Popescu - N. Popescu , Sur la definition des prolongements residuels transcendents d’une valuation sur un corps K á K(X) , Bull. Math. Soc. Sci. Math. R. S. Romania , 33 ( 81 ), No. 3 ( 1989 ), pp. 257 - 264 . MR 1043994 | Zbl 0698.12020 · Zbl 0698.12020
[8] N. Popescu - A. Zaharescu , On the structure of the irreducible polynomials over local fields , J. Number Theory , 52 , No. 1 ( 1995 ), pp. 98 - 118 . MR 1331768 | Zbl 0838.11078 · Zbl 0838.11078 · doi:10.1006/jnth.1995.1058
[9] N. Popescu - A. Zaharescu , On the main invariant of an element over a local field , Portugaliae Mathematica , 54 , Fasc. 1 ( 1997 ), pp. 73 - 83 . MR 1440129 | Zbl 0894.11045 · Zbl 0894.11045
[10] I.P. Serre , Corps locaux , Hermann , Paris ( 1962 ). Zbl 0137.02601 · Zbl 0137.02601
[11] O.F.G. Shilling , The theory of valuations , Amer. Math. Soc., Mathematical Surveys , Number IV ( 1950 ). Zbl 0037.30702 · Zbl 0037.30702
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.