Class groups of imaginary function fields: The inert case

Authors:
Yoonjin Lee and Allison M. Pacelli

Journal:
Proc. Amer. Math. Soc. **133** (2005), 2883-2889

MSC (2000):
Primary 11R29; Secondary 11R58

Published electronically:
April 22, 2005

MathSciNet review:
2159765

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite field and a transcendental element over . An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers and , there are infinitely many function fields of fixed degree such that the class group of contains a subgroup isomorphic to and the prime at infinity is inert.

**1.**Takashi Azuhata and Humio Ichimura,*On the divisibility problem of the class numbers of algebraic number fields*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**30**(1984), no. 3, 579–585. MR**731519****2.**Christian Friesen,*Class number divisibility in real quadratic function fields*, Canad. Math. Bull.**35**(1992), no. 3, 361–370. MR**1184013**, 10.4153/CMB-1992-048-5**3.**A. Frohlich and M.J. Talyor,*Algebraic Number Theory*, Cambridrige University Press, 1993.**4.**Serge Lang,*Algebra*, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR**783636****5.**T. Nagell,*Uber die Klassenzahl imaginar quadratischer Zahlkorper*, Abh. Math. Sem. Univ. Hamburg**1**(1922), 140-150.**6.**Shin Nakano,*On ideal class groups of algebraic number fields*, J. Reine Angew. Math.**358**(1985), 61–75. MR**797674**, 10.1515/crll.1985.358.61**7.**Allison M. Pacelli,*Abelian subgroups of any order in class groups of global function fields*, J. Number Theory**106**(2004), no. 1, 26–49. MR**2029780**, 10.1016/j.jnt.2003.12.003**8.**Michael Rosen,*Number theory in function fields*, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002. MR**1876657****9.**Yoshihiko Yamamoto,*On unramified Galois extensions of quadratic number fields*, Osaka J. Math.**7**(1970), 57–76. MR**0266898**

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Additional Information

**Yoonjin Lee**

Affiliation:
Department of Mathematics, Smith College, Northampton, Massachusetts 01063

Email:
yjlee@smith.edu

**Allison M. Pacelli**

Affiliation:
Department of Mathematics, Williams College, Williamstown, Massachusetts 01267

Email:
Allison.Pacelli@williams.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-07910-4

Keywords:
Class group,
class number,
rank of class group,
imaginary function field

Received by editor(s):
May 1, 2004

Received by editor(s) in revised form:
June 8, 2004

Published electronically:
April 22, 2005

Communicated by:
Wen-Ching Winnie Li

Article copyright:
© Copyright 2005
American Mathematical Society