Geodesic
Explicit constructions of extensions of complete fields of characteristic $0$
Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 179-196.

Voir la notice de l'article provenant de la source Math-Net.Ru

This survey article is devoted to $p$-extensions of complete discrete valuation fields of mixed characteristic where $p$ is the characteristic of the residue field. It is known that any totally ramified Galois extension with a non-maximal ramification jump can be determined by an Artin-Schreier equation, and the upper bound for the ramification jump corresponds to the lower bound of the valuation in the right-hand side of the equation. The problem of construction of extensions with arbitrary Galois groups is not solved.
Keywords: discrete valuation field, ramification jump, Artin-Schreier equation. 
@article{CHEB_2023_24_2_a8,
     author = {I. B. Zhukov and O. Yu. Ivanova},
     title = {Explicit constructions of extensions of complete fields of characteristic $0$},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {179--196},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_2_a8/}
}
TY  - JOUR
AU  - I. B. Zhukov
AU  - O. Yu. Ivanova
TI  - Explicit constructions of extensions of complete fields of characteristic $0$
JO  - Čebyševskij sbornik
PY  - 2023
SP  - 179
EP  - 196
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2023_24_2_a8/
LA  - ru
ID  - CHEB_2023_24_2_a8
ER  - 
%0 Journal Article
%A I. B. Zhukov
%A O. Yu. Ivanova
%T Explicit constructions of extensions of complete fields of characteristic $0$
%J Čebyševskij sbornik
%D 2023
%P 179-196
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2023_24_2_a8/
%G ru
%F CHEB_2023_24_2_a8
I. B. Zhukov; O. Yu. Ivanova. Explicit constructions of extensions of complete fields of characteristic $0$. Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 179-196. http://geodesic.mathdoc.fr/item/CHEB_2023_24_2_a8/

[1] Boitsov, V. G., Zhukov, I. B., “Continuability of cyclic extensions of complete discrete valuation fields”, J. Math. Sci. (N. Y.), 130:3 (2005), 4643–4650 | DOI | MR | Zbl

[2] Vostokov, S. V., “Explicit construction of class field theory for a multidimensional local field”, Math. USSR-Izv., 26:2 (1986), 263–287 | DOI | MR | Zbl

[3] Fesenko, I. B., Vostokov, S. V., Zhukov I. B., “On the theory of multidimensional local fields. Methods and constructions”, Algebra i Analiz, 2:4 (1990), 91–118 | Zbl

[4] Vostokov, S. V., Zhukov, I. B., “Some approaches to the construction of abelian extensions for $\mathfrak p$-adic fields”, Proceedings of the St. Petersburg Mathematical Society, v. III, Amer. Math. Soc. Transl. Ser. 2, 166, Amer. Math. Soc., Providence, RI, 1995, 157–174 | MR | Zbl

[5] Vostokov, S. V., Zhukov, I. B., Pak, G.K., “Extensions with almost maximal depth of ramification”, J. Math. Sci., 112:3 (1999), 4285–4302 | DOI | MR | Zbl

[6] Vostokov, S. V., Zhukov, I. B., Ivanova, O. Yu., “Inaba extensions of complete fields of characteristic 0”, Chebyshevskii sbornik, 20:3 (2019), 124–133 | DOI | MR | Zbl

[7] Zhukov, I. B., “Structure theorems for complete fields”, Proceedings of the St. Petersburg Mathematical Society, v. III, Amer. Math. Soc. Transl. Ser. 2, 166, Amer. Math. Soc., Providence, RI, 1995, 175–192 | MR | Zbl

[8] Zhukov, I. B., Lysenko, E. F., “Construction of cyclic extensions of degree $p^2$ for a complete field”, J. Math. Sci., 234:2 (2018), 148–157 | DOI | MR | Zbl

[9] Zhukov I. B., Ivanova O. Yu., “On Inaba extensions for two-dimensional local fields of mixed characteristic”, J. Math. Sci., 2022 | MR

[10] Zhukov I. B., Ivanova, O. Yu., Elimination of maximal jumps, In preparation

[11] Ivanova, O. Yu., “Construction of a cyclic ferocious extension by means of an Inaba equation”, J. Math. Sci., 2022

[12] Iwasawa, K., Local class field theory, Oxford University Press, 1986 | MR | MR | Zbl

[13] Cassels J. W. S., Frohlich A. (eds.), Algebraic Number Theory, Academic Press, London–New York, 1967 | MR | Zbl

[14] Lang S., Algebra, Rev. 3rd edition, Springer, New York et al., 2002 | MR | Zbl

[15] Parshin A., “Local class field theory”, Proc. Steklov Inst. Math., 165:1985, 157–185 | MR | Zbl

[16] I. B. Fesenko, S. V. Vostokov, Local fields and their extensions. A constructive approach, Second edition, AMS, Providence, RI, 2002 | MR

[17] O. Hyodo, “Wild ramification in the imperfect residue field case”, Adv. Stud. Pure Math., 12 (1987), 287–314 | DOI | MR | Zbl

[18] K. İ. Ikeda, E. Serbest, “Local abelian Kato-Parshin reciprocity law: a survey”, Hacet J. Math. Stat., 50:5 (2021), 1225–1250 | DOI | MR | Zbl

[19] E. Inaba, “On matrix equations for Galois extensions of fields with characteristic p”, Natur. Sci. Rep. Ochanomizu Univ., 12 (1961), 26–36 | MR | Zbl

[20] K. Kato, “Vanishing cycles, ramification of valuations, and class field theory”, Duke Math. J., 55 (1987), 629–659 | MR | Zbl

[21] K. Keating, P. Schwartz, “Galois scaffolds and Galois module structure for totally ramified $C_{p^2}$-extensions in characteristic 0”, Number Theory, 239 (2022), 113–136 | DOI | MR | Zbl

[22] J. Lubin, J. Tate, “Formal complex multiplication in local fields”, Ann. Math., 81 (1965), 380–387 | DOI | MR | Zbl

[23] R. E. MacKenzie, G. Whaples, “Artin-Schreier equations in characteristic zero”, Amer. J. Math., 78 (1956), 473–485 | DOI | MR | Zbl

[24] H. Miki, “On $Z_p$-extensions of complete p-adic power series fields and function fields”, J. Fac. Sci. Univ. Tokyo, Sect 1A, 21 (1974), 377–393 | MR | Zbl

[25] J.-P. Serre, Local fields, Springer, 1979 | MR | Zbl

[26] St. Petersbg. Math. J., 26:5 (2014), 695-674

[27] L. Xiao, I. Zhukov, “Ramification in the imperfect residue field case, approaches and questions”, Valuation theory in interaction, Proceedings of the 2nd international conference and workshop on valuation theory (Segovia and El Escorial, Spain, July 18-29, 2011), European Mathematical Society (EMS), Zürich, 2014, 600–656 | DOI | MR | Zbl

[28] I. Zhukov, “Explicit abelian extensions of complete discrete valuation fields”, Invitation to Higher Local Fields, Geometry and Topology Monographs, 3, eds. Fesenko, I., Kurihara, M., 2000, 117–122 | DOI | MR | Zbl