Akram Lbekkouri     (10507 Casa-Bandoeng, 20002 Casablanca, Morocco)


The paper deals with some aspects of general local fields and tries to elucidate some obscure facts. Indeed, several questions remain open, in this domain of research, and literature is getting scarce. Broadly speaking, we present a full description of the absolute Galois group in all cases with answers on the solvability, prosolvability and procyclicity. Furthermore, we give a result that makes "some'' generalization to Abhyankar's Lemma in local case. Half-way a short section, containing a view of some future research loosely discussed, presents an attempt in the development of the theory. An Annexe elucidate several important points, concerning Hilbert's theory.


Inertia group, Abhyankar's Lemma, Imperfect residue field, Weakly unramified, Solvability, Monogenity

Full Text:



  1. Abbes A., Saito T. Ramification of local fields with imperfect residue fields. Amer. J. Math., 2002. Vol. 124, No. 5, P. 879–920.
  2. Abrashkin V.A. Towards Explicit Description of Ramification Filtration in the 2-dimensional Case. Prepint of Nottingham Univ., 2000. No. 00-01.
  3. Borger J. A monogenic Hasse–Arf theorem. In: Proc. of the Conf. on Ramification Theory for Arithmetic Schemes, Luminy, 1999.
  4. Bosch S., Lütkebohmert W., Raynaud M. Néron Models. Ergeb. Math. Grenzgeb. (3), vol. 21. Berlin, Heidelberg: Springer–Verlag, 1990. 328 p. DOI: 10.1007/978-3-642-51438-8
  5. Engler A.J., Prestel A. Valued Fields. Springer Monogr. Math. Berlin, Heidelberg: Springer–Verlag, 2005. 208 p. DOI: 10.1007/3-540-30035-X
  6. Epp H.P. Eliminating wild ramification. Invent Math., 1973. Vol. 19. P. 235–249. DOI: 10.1007/BF01390208
  7. Gold R., Madan M.L. Some applications of Abhyankar’s Lemma. Math. Nachr., 1978. Vol. 82, No. 1. P. 115–119. DOI: 10.1002/mana.19780820112
  8. Jonah D., Konvisser M. Some nonabelian \(p\)-groups with abelian automorphism groups. Arch. Math., 1975. Vol. 2, No. 1. P. 131—133. DOI: 10.1007/BF01229715
  9. Koenigsmann J. Solvable absolute Galois groups are metabelian. Invent. Math., 2001. Vol. 144. P. 1–22. DOI: 10.1007/s002220000117
  10. Kuhlmann F.V. A Correction to Epp’s paper “Elimination of Wild Ramification”, 2010. arXiv: 1003.5687v1 [math.AC]
  11. Lbekkouri A. On the solvability in local extensions. An. Şt. Univ. Ovidius Constanţa, 2014. Vol. 22. No. 2. P. 121–127. DOI: 10.2478/auom-2014-0037
  12. Neukirch J., Shmidt A., Wingberg K. Cohomology of Number Fields. Grundlehren Math. Wiss., vol. 323. Berlin: Springer–Verlag, 2000. 720 p.
  13. Neukirch J. Algebraic Number Theory. Berlin, Heidelberg: Springer–Verlag, 1999. 322 p. DOI: 10.1007/978-3-662-03983-0
  14. Ribes L., Zalesskii P. Profinite Groups. Ergeb. Math. Grenzgeb. (3), vol. 40. Berlin, Heidelberg: Springer–Verlag, 2000. 483 p. DOI: 10.1007/978-3-642-01642-4
  15. Safarevič I.R. On p-extensions. Amer. Math. Soc. Transl. Ser. 2, 1954. Vol. 4. P. 59–72.
  16. Saito T. Ramification of local fields with imperfect residue fields III. Math. Ann., 2012. Vol. 352. P. 567–580. DOI: 10.1007/s00208-011-0652-5
  17. Saito T. Wild ramification and the characteristic cycle of an \(l\)-adic sheaf. J. Inst. Math. Jussieu, 2008. Vol. 8, No. 4. P. 769–829. DOI: 10.1017/S1474748008000364
  18. Serre J.-P. Local Fields. Grad. Texts in Math., vol. 67. New York: Springer–Verlag, 1979. 241 p. DOI: 10.1007/978-1-4757-5673-9
  19. Serre J.-P. Cohomologie Galoisienne. Lecture Notes in Math., vol. 5. Berlin Heidelberg: Springer–Verlag, 1997. 181 p. DOI: 10.1007/BFb0108758
  20. Spriano L. Well ramified extensions of complete discrete valuation fields with application to the Kato Conductor. Canad. J. Math., 2000. Vol. 52, No. 6. P. 1269–1309. DOI: 10.4153/CJM-2000-053-1
  21. Spriano L. On ramification theory of monogenic extensions. In: Geom. Topol. Monogr. Vol. 3: Invitation to Higher Local Fields, eds. I. Fesenko and M. Kurihara, 2000. Part I, Sect. 18. P. 151–164.
  22. Ware R. On Galois groups of maximal \(p\)-extension. Trans. Amer. Math. Soc., 1992. Vol. 333, No. 2. P. 721–728. DOI: 10.2307/2154057
  23. Xiao L. On ramification filtrations and \(p\)-adic differential modules, I: the equal characteristic case. Algebra Number Theory, 2010. Vol. 4, No. 8. P. 969–1027. DOI: 10.2140/ant.2010.4.969
  24. Xiao L. On ramification filtrations and p-adic differential equations, II: mixed characteristic case. Compos. Math., 2012. Vol. 148, No. 2. P. 415–463. DOI: 10.1112/S0010437X1100707X
  25. Zariski O., Samuel P. Commutative Algebra I. Grad. Texts in Math., vol. 28. New York: Springer–Verlag, 1975. 334 p.
  26. Zhukov I.B. On Ramification Theory in the Imperfect Residue Field Case. Prepint No. 98-02, Nottingham Univ., 1998. Accessible on arXiv: math/0201238[math.NT]
  27. Zhukov I.B. Ramification of Surfaces Artin-Schreier Extensions, 2002. arXiv: math/0209183[math.AG]
  28. Zhukov I.B. Ramification of Surfaces: Sufficient Jet Order for Wild Jumps, 2002. arXiv: math/0201071[math.AG]

DOI: http://dx.doi.org/10.15826/umj.2019.2.004

Article Metrics

Metrics Loading ...


  • There are currently no refbacks.