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Solution of small class number problems for cyclotomic fields

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Solution of small class number problems for cyclotomic fields COMPOSITIO MATHEMATICA JOHNMYRONMASLEY Solution of small class number problems for cyclotomic fields Compositio Mathematica, tome 33, no 2 (1976), p. 179-186. <http://www.numdam.org/item?id=CM_1976__33_2_179_0> © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 179 SOLUTION OF SMALL CLASS NUMBER PROBLEMS FOR CYCLOTOMIC FIELDS John Myron Masley COMPOSITIO MATHEMATICA, Vol. 33, Fasc. 2, 1976, pag. 179-186 Noordhoff International Publishing Printed in the Netherlands By an algebraic number field F we shall mean a finite extension of Q, the field of rational numbers. The class number hF of F is the order of C,, the group of ideal classes of F. In previous work ([5], [6], [7]) we have determined all F of the form F== Cm = 0(exp 21ri/m) such that hm = hCm :5 2. In this paper we extend our results and give a complete list of cyclotomic fields whose class number is less than 11. Since for m an odd integer we have Cm = C2m, we assume throughout this paper that m ~ 2 mod 4 and our result is MAIN THEOREM: Let m be an integer greater than 2, m ~ 2 mod 4. Then all the values of m for which the cyclotomic field Cm has class number hm with 2:5 hm ~ 10 are listed in the table: Furthermore, all the other values of m with ~(m) = |Cm : Q| ~ 24 give the twenty-nine values of m for which hm = 1. In § 1 we use group actions on the ideal class group to prove a general lemma which is of interest in itself. Essentia

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