On the Stickelberger ideal and the relative class number

Authors:
Tatsuo Kimura and Kuniaki Horie

Journal:
Trans. Amer. Math. Soc. **302** (1987), 727-739

MSC:
Primary 11R18; Secondary 11R29

DOI:
https://doi.org/10.1090/S0002-9947-1987-0891643-8

MathSciNet review:
891643

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Abstract | References | Similar Articles | Additional Information

Abstract: Let $k$ be any imaginary abelian field, $R$ the integral group ring of $G = {\text {Gal}}(k/\mathbb {Q})$, and $S$ the Stickelberger ideal of $k$. Roughly speaking, the relative class number ${h^ - }$ of $k$ is expressed as the index of $S$ in a certain ideal $A$ of $R$ described by means of $G$ and the complex conjugation of $k;{c^ - }{h^ - } = [A:S]$, with a rational number ${c^ - }$ in $\frac {1} {2}\mathbb {N} = \{ n/2;n \in \mathbb {N}\}$, which can be described without ${h^ - }$ and is of lower than ${h^ - }$ if the conductor of $k$ is sufficiently large (cf. [**6, 9, 10**]; see also [**5**]). We shall prove that $2{c^ - }$, a natural number, divides $2{([k:\mathbb {Q}]/2)^{[k:\mathbb {Q}]/2}}$. In particular, if $k$ varies through a sequence of imaginary abelian fields of degrees bounded, then ${c^ - }$ takes only a finite number of values. On the other hand, it will be shown that ${c^ - }$ can take any value in $\frac {1} {2}\mathbb {N}$ when $k$ ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.

- Gary Cornell,
*Abhyankar’s lemma and the class group*, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 82–88. MR**564924** - Gary Cornell and Lawrence C. Washington,
*Class numbers of cyclotomic fields*, J. Number Theory**21**(1985), no. 3, 260–274. MR**814005**, DOI https://doi.org/10.1016/0022-314X%2885%2990055-1 - Frank Gerth III,
*Asymptotic results for class number divisibility in cyclotomic fields*, Canad. Math. Bull.**26**(1983), no. 4, 464–472. MR**716587**, DOI https://doi.org/10.4153/CMB-1983-075-3 - Helmut Hasse,
*Über die Klassenzahl abelscher Zahlkörper*, Akademie-Verlag, Berlin, 1952 (German). MR**0049239** - Kuniaki Horie,
*On the index of the Stickelberger ideal and the cyclotomic regulator*, J. Number Theory**20**(1985), no. 2, 238–253. MR**790784**, DOI https://doi.org/10.1016/0022-314X%2885%2990042-3 - Kenkichi Iwasawa,
*A class number formula for cyclotomic fields*, Ann. of Math. (2)**76**(1962), 171–179. MR**154862**, DOI https://doi.org/10.2307/1970270 - Tatsuo Kimura and Kuniaki Horie,
*On the Stickelberger ideal and the relative class number*, Proc. Japan Acad. Ser. A Math. Sci.**58**(1982), no. 4, 170–171. MR**664565** - J. Myron Masley and Hugh L. Montgomery,
*Cyclotomic fields with unique factorization*, J. Reine Angew. Math.**286(287)**(1976), 248–256. MR**429824** - W. Sinnott,
*On the Stickelberger ideal and the circular units of a cyclotomic field*, Ann. of Math. (2)**108**(1978), no. 1, 107–134. MR**485778**, DOI https://doi.org/10.2307/1970932 - W. Sinnott,
*On the Stickelberger ideal and the circular units of an abelian field*, Invent. Math.**62**(1980/81), no. 2, 181–234. MR**595586**, DOI https://doi.org/10.1007/BF01389158

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Keywords:
(Imaginary) abelian field,
Stickelberger ideal,
relative class number,
analytic class number formula

Article copyright:
© Copyright 1987
American Mathematical Society