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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. <> © Foundation Compositio Mathematica, 1976, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( 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 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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