Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
discrete valuation; extension of valuation; prime ideal factorization
discrete valuation; extension of valuation; prime ideal factorization
Summary:
Let $(K,\nu)$ be a valued field, where $\nu$ is a rank one discrete valuation. Let $R$ be its ring of valuation, ${\mathfrak m}$ its maximal ideal, and $L$ an extension of $K$, defined by a monic irreducible polynomial $F(X) \in R[X]$. Assume that $\overline{F}(X)$ factors as a product of $r$ distinct powers of monic irreducible polynomials. In this paper a condition which guarantees the existence of exactly $r$ distinct valuations of $K$ extending $\nu$ is given, in such a way that it generalizes the results given in the paper ``Prolongations of valuations to finite extensions" by S.\,K. Khanduja, M. Kumar (2010).
References:
[1] Cohen S. D., Movahhedi A., Salinier A.: Factorization over local fields and the irreducibility of generalized difference polynomials. Mathematika 47 (2000), no. 1–2, 173–196. DOI 10.1112/S0025579300015801 | MR 1924496
[2] Deajim A., El Fadil L.: On the extensions of a discrete valuation in a number field. Math. Slovaca 69 (2019), no. 5, 1009–1022. DOI 10.1515/ms-2017-0285 | MR 4017386
[3] Dedekind R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen 23 (1878), 3–38 (German).
[4] Guàrdia J., Montes J., Nart E.: Newton polygons of higher order in algebraic number theory. Trans. Amer. Math. Soc. 364 (2012), no. 1, 361–416. DOI 10.1090/S0002-9947-2011-05442-5 | MR 2833586
[5] Hensel K.: Untersuchung der Fundamentalgleichung einer Gattung für eine reelle Primzahl als Modul und Bestimmung der Theiler ihrer Discriminante. J. Reine Angew. Math. 113 (1894), 61–83 (German). MR 1580345
[6] Khanduja S. K., Kumar M.: On a theorem of Dedekind. Int. J. Number Theory 4 (2008), no. 6, 1019–1025. DOI 10.1142/S1793042108001833 | MR 2483309
[7] Khanduja S. K., Kumar M.: Prolongations of valuations to finite extensions. Manuscripta Math. 131 (2010), no. 3–4, 323–334. DOI 10.1007/s00229-009-0320-1 | MR 2592083
[8] Khanduja S. K., Kumar M.: A generalization of a theorem of Ore. J. Pure Appl. Algebra 218 (2014), no. 7, 1206–1218. DOI 10.1016/j.jpaa.2013.11.014 | MR 3168492
[9] Neukirch J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften, 322, Springer, Berlin, 1999. DOI 10.1007/978-3-662-03983-0 | MR 1697859
[10] Ore Ö.: Newtonsche Polygone in der Theorie der algebraischen Körper. Math. Ann. 99 (1928), no. 1, 84–117 (German). DOI 10.1007/BF01459087 | MR 1512440
Search
Partner of
