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Notes on the structure of the ideal class groups of abelian number fields. (English) Zbl 1084.11060

Euler systems of cyclotomic units and Stickelberger elements introduced by V. A. Kolyvagin [Grothendieck Festschrift 435–483 (1990; Zbl 0742.14017)] and L. C. Washington [Introduction to Cyclotomic Fields. 2nd ed. Graduate Texts in Mathematics. 83. (New York, NY: Springer) (1997; Zbl 0966.11047)] turned out to be an important tool for studying the structure of the ideal class groups of cyclotomic number fields.
In this article, the author derives explicit formulas for certain higher annihilators, and uses them to derive results on the structure of the \(p\)-class groups of cyclotomic number fields of degree not divisible by \(p\).

MSC:

11R29 Class numbers, class groups, discriminants
11R20 Other abelian and metabelian extensions
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References:

[1] V. A. Kolyvagin, Euler systems, in The Grothendieck Festschrift, Vol. II , Progr. Math., 87, Birkhäuser Boston, Boston, 1990, pp. 435-483. · Zbl 0742.14017
[2] B. Mazur and A. Wiles, Class fields of abelian extensions of \(\textbf{Q}\), Invent. Math. 76 (1984), no. 2, 179-330. · Zbl 0545.12005 · doi:10.1007/BF01388599
[3] S. Lang, Cyclotomic fields I and II , Combined 2nd ed., Springer, New York, 1990, pp. 397-419.
[4] K. Rubin, Kolyvagin’s system of Gauss sums, in Arithmetic algebraic geometry ( Texel, 1989 ) , 309-324 Progr. Math., 89, Birkhäuser Boston, Boston, 1991. · Zbl 0727.11044 · doi:10.1007/978-1-4612-0457-2_14
[5] F. Thaine, On the ideal class groups of real abelian number fields, Ann. of Math. (2) 128 (1988), no. 1, 1-18. · Zbl 0665.12003 · doi:10.2307/1971460
[6] L. C. Washington, Introduction to cyclotomic fields , 2nd ed., Springer, New York, 1997. · Zbl 0966.11047
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