# zbMATH — the first resource for mathematics

Some bases of the Stickelberger ideal. (English) Zbl 0798.11044
Let $$I$$ denote the Stickelberger ideal of the group ring $$R = \mathbb{Z} [\text{Gal} (K/ \mathbb{Q})]$$, where $$K$$ is the $$p^{n+1}$$-th cyclotomic field $$(p$$ an odd prime, $$n \geq 0)$$. The author studies various bases of the $$\mathbb{Z}$$-module $$I$$. As an application he provides a new proof for Sinnott’s formula $$[R^*:I] = h^ -$$, where $$R^*$$ is a certain subring of $$R$$ and $$h^ -$$ denotes the relative class number of $$K$$ [see W. Sinnott, Invent. Math. 62, 181-234 (1980; Zbl 0465.12001)]. Moreover he gives both a $$\pi$$-adic version $$(\pi$$ any rational prime) and a modulo $$p$$ version of this index formula. The latter reads $$[R^* (p) : I(p)] = p^{i(p)}$$, where $$R^* (p)$$ and $$I(p)$$ are reductions mod $$p$$ of $$R^*$$ and $$I$$, respectively, and $$i(p)$$ denotes the index of irregularity of $$p$$.
As a further application the author proves the equivalence of several systems of congruences mod $$p$$ introduced by various authors in the study of the Fermat equation $$x^ p + y^ p = z^ p$$. One of these systems is the well-known Kummer system involving Bernoulli numbers and Mirimanoff polynomials. In this piece of work the author continues his earlier study [see Comment. Math. Univ. St. Pauli 31, 89-97 (1982; Zbl 0496.10006)].

##### MSC:
 11R18 Cyclotomic extensions 11D41 Higher degree equations; Fermat’s equation 11R29 Class numbers, class groups, discriminants 16S34 Group rings
Full Text:
##### References:
 [1] AGOH T.: On Fermat’s last theorem. C.R. Math. Rep. Acad. Sci. Canada VII (1990), 11-15. · Zbl 0703.11015 [2] BENNETON G.: Sur le dernier théorème de Fermat. Ann. Sci. Univ. Besançon Math. 3 (1974), 15pp. · Zbl 0348.10010 [3] FUETER R.: Kummers Kriterium zum letzten Theorem von Fermat. Math. Ann. 85 (1922), 11-20. · JFM 48.0130.05 [4] GRANVILLE A. J.: Diophantine Equations with Varying Exponents (with Special Reference to Fermat’s Last Theorem). Ph. D. thesis, Queen’s University, 1987. [5] IWASAWA K.: A class number formula for cyclotomic fields. Ann. of Math. 76 (1962), 171-179. · Zbl 0125.02003 [6] KUČERA R.: On bases of the Stickelberger ideal and of the group of circular units of a cyclotomic fields. J. Number Theory 40 (1992), 284-316. · Zbl 0744.11052 [7] KUMMER E. E.: Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren. J. Reine Angew. Math. 35 (1847), 327-367) · ERAM 035.0993cj [8] KUMMER E. E.: Einige Sëtze über die aus den Wurzeln der Gleichung \alpha \lambda = 1 gebildeten complexen Zahlen, für den Fall, daß die Klassenanzahl durch \lambda theilbar ist, nebst Anwendung derselben auf einen weiteren Beweis des letzten Fermat’schen Lehrsatzes. Abh. Königl. Akad. Wiss., Berlin (1857), 41-74) [9] LERCH M.: Zur Theorie des Fermatschen Quotienten (a p-1 - 1)/p = q(a). Math. Ann. 60 (1905), 471-490. · JFM 36.0266.03 [10] LE LIDEC P.: Sur une forme nouvelle des congruences de Kummer-Mirimanoff. C.R. Acad. Sci. Paris Sér. A 265 (1967), 89-90. · Zbl 0154.29602 [11] LE LIDEC P.: Nouvelle forme des congruences de Kummer-Mirimanoff pour le premier cas du théorème de Fermat. Bull. Soc. Math. France 97 (1969), 321-328. · Zbl 0188.10102 [12] NEWMAN M.: A table of the first factor for prime cyclotomic fields. Math. Comp. 24(109) (1970), 215-219. · Zbl 0198.36902 [13] SINNOTT W.: On the Stickelberger ideal and the circular units of a cyclotomic field. Ann. of Math. 108 (1978), 107-134. · Zbl 0395.12014 [14] SINNOTT W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math. 62, Springer, Berlin-New York, 1980, pp. 181-234. · Zbl 0465.12001 [15] SKULA L.: Index of irregularity of a prime. J. Reine Angew. Math. 315 (1980), 92-106. · Zbl 0419.10016 [16] SKULA L.: Another proof of Iwasawa’s class number formula. Acta Arith. XXXIX (1981), 1-6. · Zbl 0372.12012 [17] SKULA L.: A remark on Mirimanoff polynomials. Comment. Math. Univ. St. Paul. 31 (1982), 89-97. · Zbl 0496.10006 [18] SKULA L.: Systems of equations depending on certain ideals. Arch. Math. (Brno) 21 (1985), 23-38. · Zbl 0589.12005 [19] SKULA L.: A note on the index of irregularity. J. Number Theory 22 (1986), 125-138. · Zbl 0589.12006 [20] VANDIVER H. S.: A property of cyclotomic integers and its relation to Fermat’s last theorem. Ann. of Math. 21 (1919-20), 73-80. · JFM 47.0151.03 [21] WASHINGTON L. C.: Introduction to Cyclotomic Fields. Springer-Verlag, New York-Heidelberg-Berlin, 1982. · Zbl 0484.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.