Explicit constructions of extensions of complete fields of characteristic 0
https://doi.org/10.22405/2226-8383-2023-24-2-179-196
Abstract
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.
About the Authors
Igor Borisovich Zhukov
Saint Petersburg State University
Russian Federation
Russian Federation
Olga Yur’evna Ivanova
Saint Petersburg State University
Russian Federation
Russian Federation
References
1. Boitsov, V. G., Zhukov, I. B. 2005 “Continuability of cyclic extensions of complete discrete valuation fields”, J. Math. Sci. (N. Y.), vol. 130, № 3 (2005), pp. 4643-4650.
2. Vostokov, S. V. 1986 “Explicit construction of class field theory for a multidimensional local field” Math. USSR-Izv., vol. 26, № 2, pp. 263—287.
3. Fesenko, I. B., Vostokov, S. V., Zhukov I. B. 1990, “On the theory of multidimensional local fields. Methods and constructions”, Algebra i Analiz vol. 2, № 4. pp. 91-118.
4. Vostokov, S. V., Zhukov, I. B. 1995, “Some approaches to the construction of abelian extensions for p-adic fields”, Proceedings of the St. Petersburg Mathematical Society, vol. III, pp. 157–174, Amer. Math. Soc. Transl. Ser. 2, 166, Amer. Math. Soc., Providence, RI, 1995.
5. Vostokov, S. V., Zhukov, I. B., Pak, G.K. 1999, “Extensions with almost maximal depth of ramification”, J. Math. Sci., vol. 112, № 3, pp. 4285-4302
6. Vostokov, S. V., Zhukov, I. B., Ivanova, O. Yu. 2019 “Inaba extensions of complete fields of characteristic 0” Chebyshevskii sbornik vol. 20, № 3, pp. 124-133
7. Zhukov, I. B., 1995 “Structure theorems for complete fields”, Proceedings of the St. Petersburg Mathematical Society, vol. III, pp. 175–192, Amer. Math. Soc. Transl. Ser. 2, 166, Amer. Math. Soc., Providence, RI, 1995.
8. Zhukov, I.B., Lysenko, E. F. 2018, “Construction of cyclic extensions of degree 𝑝2 for a complete field”, J. Math. Sci., vol. 234(2), pp. 148–157
9. Zhukov I. B., Ivanova, O. Yu. 2022 “On Inaba extensions for two-dimensional local fields of mixed characteristic” J. Math. Sci., to appear.
10. Zhukov I. B., Ivanova, O. Yu. “Elimination of maximal jumps” In preparation.
11. Ivanova, O. Yu. 2022 “Construction of a cyclic ferocious extension by means of an Inaba equation” J. Math. Sci., to appear.
12. Iwasawa, K. 1986 Local class field theory, Oxford University Press.
13. Cassels, J. W. S. Frohlich, A. (eds.), 1967, “Algebraic Number Theory”, Academic Press, London and New York.
14. Lang. S. 2002 “Algebra”, Rev. 3rd edition, Springer, New York et al.
15. Parshin, A. N. 1985 “Local class field theory” Proc. Steklov Inst. Math., vol. 165, pp. 157-185.
16. Fesenko I. B., Vostokov, S. V., 2002, “Local fields and their extensions. A constructive approach”,
17. Second edition, AMS, Providence, RI.
18. Hyodo, O., 1987, “Wild ramification in the imperfect residue field case”, Adv. Stud. Pure Math., vol. 12, pp. 287-314.
19. Ikeda, K. ˙I., Serbest E. 2021, “Local abelian Kato-Parshin reciprocity law: a survey”, K. ˙I. Ikeda, E. Serbest Hacet J. Math. Stat., vol. 50, № 5, pp. 1225-1250.
20. Inaba, E., 1961, “On matrix equations for Galois extensions of fields with characteristic 𝑝”, Natur. Sci. Rep. Ochanomizu Univ., vol. 12, pp. 26-36.
21. Kato, K. 1987 “Vanishing cycles, ramification of valuations, and class field theory” Duke Math. J., vol. 55, pp. 629-659.
22. Keating, K. and Schwartz, P., 2022 “Galois scaffolds and Galois module structure for totally ramified 𝐶𝑝2-extensions in characteristic 0” J. Number Theory, vol. 239, pp. 113-136.
23. Lubin, J., Tate, J. 1965 “Formal complex multiplication in local fields”, Ann. Math. vol. 81, pp. 380-387.
24. MacKenzie, R. E., Whaples, G., 1956, “Artin–Schreier equations in characteristic zero”, Amer. J. Math., vol. 78, pp. 473-485.
25. Miki, H. 1974, “On 𝑍𝑝-extensions of complete p-adic power series fields and function fields”, J. Fac. Sci. Univ. Tokyo, Sect 1A, vol. 21, pp. 377-393.
26. Serre, J.-P. 1979, “Local fields”, Springer.
27. Xiao, L., Zhukov, I. 2014, “Ramification in the imperfect residue field case, approaches and questions”, Algebra i Analiz, vol. 26, № 5. pp. 695-740. St. Petersbg. Math. J. 26, № 5, pp. 695-74
28. Xiao, L., Zhukov, I. 2014 “Ramification in the imperfect residue field case, approaches and questions” L. Xiao, I. Zhukov in: 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), Z¨urich, 2014, pp. 600-656 (Zbl 1312.14010)
29. Zhukov, I., 2000, “Explicit abelian extensions of complete discrete valuation fields”, in book: Fesenko, I., Kurihara, M. (eds.) Invitation to Higher Local Fields. Geometry and Topology Monographs, vol. 3, pp. 117-122.
Review
For citations:
Zhukov I.B., Ivanova O.Yu. Explicit constructions of extensions of complete fields of characteristic 0. Chebyshevskii Sbornik. 2023;24(2):179-196. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-2-179-196
Views:
323
Email this article 