Download PDF - Determination of the imaginary normal octic number fields with class number one which are not CM-fieldsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Determination of the imaginary normal octic number fields with class number one which are not CM-fields

Authors 1

Affiliations

  1. Department of Mathematics, National Defence Academy, Hashirimizu Yokosuka 239-8686, Japan

Bibliography

  1. S. Arno, The imaginary quadratic fields of class number 4, Acta Arith. 60 (1992), 321-334; MR 93b:11144.
  2. E. Brown and C. J. Parry, The imaginary bicyclic biquadratic fields with class numbers 1, J. Reine Angew. Math. 266 (1974), 118-120; MR 49 #4974.
  3. D. A. Buell, H. C. Williams and K. S. Williams, On the imaginary bicyclic biquadratic fields with class-number 2, Math. Comp. 31 (1977), 1034-1042; MR 56 #305.
  4. C. Castela, Nombre de classes d'idéaux d'une extension diédrale de degré 8 de ℚ, Séminaire de Théorie des Nombres 1977-1978, Exp. No. 5, 8 pp., CNRS, Talence, 1978; MR 81e:12006.
  5. C. Castela, Nombre de classes d'idéaux d'une extension diédrale d'un corps de nombres, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), 483-486; MR 80c:12012.
  6. P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, World Sci. Publishing, Singapore, 1988; MR 90f:11092.
  7. J. Cougnard, Groupe des unités et nombre de classes de certaines extensions diédrales de degré 8 de ℚ, Théorie des Nombres, Besançon, 1983-1984, 21 pp., Publ. Math. Fac. Sci., Exp. No. 2, Besançon, Univ. Franche-Comté, Besançon, 1984; MR 87e:11126.
  8. G. Frey und W. D. Geyer, Über die Fundamentalgruppe von Körpern mit Divisorentheorie, J. Reine Angew. Math. 254 (1972), 110-122; MR 46 #7197.
  9. T. Funakura, On integral basis of pure quartic fields, Math. J. Okayama Univ. 26 (1984), 27-41; MR 86c:11089.
  10. K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257-258; MR 18 644e.
  11. S. Louboutin, On the class number one problem for non-normal quartic CM-fields, Tôhoku Math. J. (2) 46 (1994), no. 1, 1-12; MR 94m:11130.
  12. S. Louboutin and R. Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one, Acta Arith. 67 (1994), 47-62; MR 95g:11107.
  13. H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, ibid. 24 (1973//74), 529-542; MR 50 #9841.
  14. A. M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975), 275-286; MR 51 #12788.
  15. C. J. Parry, Pure quartic number fields whose class numbers are even, J. Reine Angew. Math. 272 (1974), 102-112; MR 51 #436.
  16. C. J. Parry, A genus theory for quartic fields, ibid. 314 (1980), 40-71; MR 81j:12002.
  17. L. Rédei und H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, ibid. 170 (1933), 69-74.
  18. H. M. Stark, On complex quadratic fields with class-number equal to one, Trans. Amer. Math. Soc. 122 (1966), 112-119; MR 33 #4043.
  19. O. Taussky, A remark on the class field tower, J. London Math. Soc. 12 (1937), 82-85.
  20. K. Uchida, Class numbers of imaginary abelian number fields, I, Tôhoku Math. J. (2) 23 (1971), 97-104; MR 44 #2727.
  21. K. Uchida, Imaginary abelian number fields with class numbers one, ibid. 24 (1972), 487-499; MR 48 #269.
  22. K. Uchida, On imaginary Galois extension fields with class number one, Sûgaku 25 (1973), 172-173 (in Japanese); MR 58 #27904.
  23. T. P. Vaughan, Constructing quaternionic fields, Glasgow Math. J. 34 (1992), 43-54; MR 92m:12005.
  24. Y. Yamamoto, Divisibility by 16 of class numbers of quadratic fields whose 2-class groups are cyclic, Osaka J. Math. 21 (1984), 1-22; MR 85g:11092.
  25. K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899-921; MR 94g:11096.
  26. K. Yamamura, Maximal unramified extensions of imaginary quadratic number fields of small conductors, J. Théor. Nombres Bordeaux 9 (1997), 405-448.
Acta Arithmetica
1998/86/2
Export to PBN, Opens in new tab
Pages:
133-147
Main language of publication
English
Received
1997-07-15
Accepted
1998-06-22
Published
1998
Exact and natural sciences