Rocky Mountain Journal of Mathematics

Genus fields of abelian extensions of rational congruence function fields, II

Jonny Fernando Barreto-Castaneda, Carlos Montelongo-Vazquez, Carlos Daniel Reyes-Morales, Martha Rzedowski-Calderon, and Gabriel Villa-Salvador

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Abstract

In this paper, we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the particular case of finite abelian $p$-extensions and give an explicit description of their genus field.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2099-2133.

Dates
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1544756803

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2099

Mathematical Reviews number (MathSciNet)
MR3892126

Zentralblatt MATH identifier
06999256

Subjects
Primary: 11R58: Arithmetic theory of algebraic function fields [See also 14-XX]
Secondary: 11R29: Class numbers, class groups, discriminants 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.)

Keywords
Global function fields ramification genus fields abelian $p$-extensions

Citation

Barreto-Castaneda, Jonny Fernando; Montelongo-Vazquez, Carlos; Reyes-Morales, Carlos Daniel; Rzedowski-Calderon, Martha; Villa-Salvador, Gabriel. Genus fields of abelian extensions of rational congruence function fields, II. Rocky Mountain J. Math. 48 (2018), no. 7, 2099--2133. doi:10.1216/RMJ-2018-48-7-2099. https://projecteuclid.org/euclid.rmjm/1544756803


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References

  • B. Anglès and J.-F. Jaulent, Théorie des genres des corps globaux, Manuscr. Math. 101 (2000), 513–532.
  • E. Artin and J. Tate, Class field theory, Benjamin, New York, 1967.
  • S. Bae and J.K. Sunghan, Genus theory for function fields, J. Australian Math. Soc. 60 (1996), 301–310.
  • J.F. Barreto-Castañeda, F. Jarquín-Zárate, M. Rzedowski-Calderón and G. Villa-Salvador, Abelian $p$-extensions and additive polynomials, Inter. J. Math. 28 (2017), 1750100-1–1750100-32.
  • V. Bautista-Ancona, M. Rzedowski-Calderón and G. Villa-Salvador, Genus fields of cyclic $l$-extensions of rational function fields, Inter. J. Num. Th. 9 (2013), 1249–1262.
  • R. Clement, The genus field of an algebraic function field, J. Num. Th. 40 (1992), 359–375.
  • A. Fröhlich, The genus field and genus group in finite number fields, Mathematika 6 (1959), 40–46.
  • ––––, The genus field and genus group in finite number fields, II, Mathematika 6 (1959), 142–146.
  • ––––, Central extensions, Galois groups and ideal class groups of number fields, Contemp. Math. 24 (1983).
  • A. Garcia and H. Stichtenoth, Elementary abelian $p$-extensions of algebraic function fields, Manuscr. Math. 72 (1991), 67–79.
  • C.F. Gauss, Disquisitiones arithmeticae, 1801.
  • H. Hasse, Zur Geschlechtertheorie in quadratischen Zahlkörpern, J. Math. Soc. Japan 3 (1951), 45–51.
  • D. Hayes, Explicit class field theory for rational function fields, Trans. Amer. Math. Soc. 189 (1974), 77–91.
  • S. Hu and Y. Li, The genus fields of Artin-Schreier extensions, Finite Fields Appl. 16 (2010), 255–264.
  • M. Ishida, The genus fields of algebraic number fields, Lect. Notes Math. 555 (1976).
  • E. Kani, Relations between the genera and between the Hasse-Witt invariants of Galois coverings of curves, Canadian Math. Bull. 28 (1985), 321–327.
  • G. Lachaud, Artin-Schreier curves, exponential sums, and the Carlitz-Uchiyama bound for geometric codes, J. Num. Th. 39 (1991), 18–40.
  • H.W. Leopoldt, Zur Geschlechtertheorie in abelschen Zahlkörpern, Math. Nachr. 9 (1953), 351–362.
  • M. Maldonado-Ramírez, M. Rzedowski-Calderón and G. Villa-Salvador, Genus fields of abelian extensions of congruence rational function fields, Finite Fields Appl. 20 (2013), 40–54.
  • ––––, Corrigendum to Genus fields of abelian extensions of rational congruence function fields, Finite Fields Appl. 20, (2015), 283–285.
  • ––––, Genus fields of congruence function fields, Finite Fields Appl. 44 (2017), 56–75.
  • O. Ore, On a special class of polynomials, Trans. Amer. Math. Soc. 35 (1933), 559–584.
  • G. Peng, The genus fields of Kummer function fields, J. Num. Th. 98 (2003), 221–227.
  • M. Rosen, The Hilbert class field in function fields, Expos. Math. 5 (1987), 365–378.
  • H.L. Schmid, Zur Arithmetik der zyklischen p-Körpe, J. reine angew. Math. 176 (1936), 161–167.
  • G.D. Villa-Salvador, Topics in the theory of algebraic function fields, Math. Th. Appl. (2006).
  • E. Witt, Zyklische Körper und Algebren der Characteristik $p$ von Grad $p^n$, J. reine angew. Math. 176 (1936), 126–140.
  • C. Wittmann, $l$-class groups of cyclic function fields of degree $l$, Finite Fields Appl. 13 (2007), 327–347.
  • X. Zhang, A simple construction of genus fields of abelian number fields, Proc. Amer. Math. Soc. 94 (1985), 393–395.