On prolongations of rank one discrete valuations

Lhoussain El Fadil

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 3, page 299-304
  • ISSN: 0010-2628

Abstract

top
Let ( K , ν ) be a valued field, where ν is a rank one discrete valuation. Let R be its ring of valuation, 𝔪 its maximal ideal, and L an extension of K , defined by a monic irreducible polynomial F ( X ) ∈ R [ X ] . Assume that 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 ν 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).

How to cite

top

El Fadil, Lhoussain. "On prolongations of rank one discrete valuations." Commentationes Mathematicae Universitatis Carolinae 60.3 (2019): 299-304. <http://eudml.org/doc/294325>.

@article{ElFadil2019,
abstract = {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).},
author = {El Fadil, Lhoussain},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {discrete valuation; extension of valuation; prime ideal factorization},
language = {eng},
number = {3},
pages = {299-304},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On prolongations of rank one discrete valuations},
url = {http://eudml.org/doc/294325},
volume = {60},
year = {2019},
}

TY - JOUR
AU - El Fadil, Lhoussain
TI - On prolongations of rank one discrete valuations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 3
SP - 299
EP - 304
AB - 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).
LA - eng
KW - discrete valuation; extension of valuation; prime ideal factorization
UR - http://eudml.org/doc/294325
ER -

References

top
  1. Cohen S. D., Movahhedi A., Salinier A., 10.1112/S0025579300015801, Mathematika 47 (2000), no. 1–2, 173–196. MR1924496DOI10.1112/S0025579300015801
  2. Deajim A., El Fadil L., 10.1515/ms-2017-0285, Math. Slovaca 69 (2019), no. 5, 1009–1022. MR4017386DOI10.1515/ms-2017-0285
  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., 10.1090/S0002-9947-2011-05442-5, Trans. Amer. Math. Soc. 364 (2012), no. 1, 361–416. MR2833586DOI10.1090/S0002-9947-2011-05442-5
  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). MR1580345
  6. Khanduja S. K., Kumar M., 10.1142/S1793042108001833, Int. J. Number Theory 4 (2008), no. 6, 1019–1025. MR2483309DOI10.1142/S1793042108001833
  7. Khanduja S. K., Kumar M., 10.1007/s00229-009-0320-1, Manuscripta Math. 131 (2010), no. 3–4, 323–334. MR2592083DOI10.1007/s00229-009-0320-1
  8. Khanduja S. K., Kumar M., 10.1016/j.jpaa.2013.11.014, J. Pure Appl. Algebra 218 (2014), no. 7, 1206–1218. MR3168492DOI10.1016/j.jpaa.2013.11.014
  9. Neukirch J., 10.1007/978-3-662-03983-0, Grundlehren der Mathematischen Wissenschaften, 322, Springer, Berlin, 1999. MR1697859DOI10.1007/978-3-662-03983-0
  10. Ore Ö., 10.1007/BF01459087, Math. Ann. 99 (1928), no. 1, 84–117 (German). MR1512440DOI10.1007/BF01459087

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.