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The class groups of the imaginary Abelian number fields with Galois group (ℤ/2ℤ)n

Published online by Cambridge University Press:  17 April 2009

Jeoung-Hwan Ahn  and
Soun-Hi Kwon
Jeoung-Hwan Ahn
Affiliation:
Department of Mathematics Education, Korea University, 136–701, Seoul, Korea e-mail: jh-ahn@korea.ac.kr, sounhikwon@korea.ac.kr
Soun-Hi Kwon
Affiliation:
Department of Mathematics Education, Korea University, 136–701, Seoul, Korea e-mail: jh-ahn@korea.ac.kr, sounhikwon@korea.ac.kr
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Assuming the Generalised Riemann Hypothesis we determine all imaginary Abelian number fields N whose Galois group G(N/ℚ) is isomorphic to (ℤ/2ℤ)n for some integers n ≥ 1 and the square of every ideal of N is principal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 267 - 277
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Batut, C., Belabas, K., Bernardi, D., Cohen, H. and Oliver, M., PARI-GP version 2.1.5.Google Scholar
[2]Borevich, Z.I. and Shafarevich, I.R., Number theory (Academic Press, New York and London, 1966).Google Scholar
[3]Boyd, D.W. and Kisilevsky, H., ‘On the exponent of the ideal class groups of complex quadratic fields’, Proc. Amer. Math. Soc. 31 (1972), 433–436.Google Scholar
[4]Cassels, J.W.S. and Fröhlich, A., Algebraic number theory (Academic Press, London, New York, 1967).Google Scholar
[5]Chang, K.-Y. and Kwon, S.-H., ‘The imaginary abelian number fields with class numbers equal to their genus class numbers’, J. Theor. Nombres Bordeaux 12 (2000), 349–365.CrossRefGoogle Scholar
[6]Chowla, S., ‘An extension of Heilbronn's class-number theorem’, Quart. J. Math. Oxford Ser. 2 5 (1934), 304–307.CrossRefGoogle Scholar
[7]Conner, P.E. and Hurrelbrink, J., ‘Class number parity’, Series in Pure Mathematics 8 (World Scientific, Singapore).Google Scholar
[8]Cox, D.A., Primes of the form x 2 + ny 2 (John Wiley & Sons, Inc., 1989).Google Scholar
[9]Daberkow, M., Fieker, C., Klüners, J., Phost, M., Roegner, K. and Wildanger, K., ‘Computational algebra and number theory, KANT V4’, J. Symbolic Comp. 24 (1997), 267–283.CrossRefGoogle Scholar
[10]Earnest, A.G., ‘Exponents of the class groups of imaginary abelian number fields’, Bull. Austral. Math. Soc. 35 (1987), 231–245.CrossRefGoogle Scholar
[11]Earnest, A.G. and Körner, O.H., ‘On ideal class groups of 2-power exponent’, Proc. Amer. Math. Soc. 36 (1982), 196–198.CrossRefGoogle Scholar
[12]Horie, K., ‘On the exponents of ideal class groups of cyclotomic fields’, Proc. Amer. Math. Soc. 119 (1993), 1049–1052.CrossRefGoogle Scholar
[13]Horie, K. and Horie, M., ‘CM-fields and exponents of their ideal class groups’, Acta Arith. 55 (1990), 157–170.CrossRefGoogle Scholar
[14]Horie, K. and Horie, M., ‘On the exponents of ideal class groups of CM-fields’, Lecture Notes in Math. 1434 (1990), 143–148.CrossRefGoogle Scholar
[15]Horie, K. and Horie, M., ‘On the 2-class groups of cyclotomic fields whose maximal real subfields have odd class numbers’, Proc. Amer. Math. Soc. 123 (1995), 2643–2649.CrossRefGoogle Scholar
[16]Louboutin, S., ‘Minoration(sous l'hypothèse de Riemann généralisée) des nombres de classes des corps quadratiques imaginaires. Application’, C. R. Acad. Sci. Paris 310 (1990), 795–800.Google Scholar
[17]Louboutin, S., ‘Determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ≤ 2’, Math. Comp. 64 (1995), 323–340.Google Scholar
[18]Louboutin, S., ‘Determination of all quaternion CM-fields with ideal class groups of exponent 2’, Osaka J. Math. 36 (1999), 229–257.Google Scholar
[19]Louboutin, S., Yang, H.-S. and Kwon, S.-H., ‘The non-normal quartic CM-fields and the dihedral octic CM-fields with ideal class group of exponent ≤ 2’, Math. Slovaca (to appear).Google Scholar
[20]Miyada, I., ‘On imaginary abelian number fields of type (2, …, 2) with one class in each genus’, Manuscripta Math. 88 (1995), 535–540.CrossRefGoogle Scholar
[21]Niven, I., Zuckerman, H.S. and Montgomery, H.L., An introduction to the theory of numbers, (fifth edition) (John Wiley & Sons, Inc., 1991).Google Scholar
[22]Pappalardi, F., ‘On the exponent of the ideal class group of ’, Proc. Amer. Math. Soc. 123 (1995), 663–671.Google Scholar
[23]Washington, L., Introduction to cyclotomic fields (2nd edition), Graduate texts in Mathematics 83 (Springer-Verlag, New York, 1997).CrossRefGoogle Scholar
[24]Weinberger, P.J., ‘Exponents of the class groups of complex quadratic fields’, Acta Arith. 22 (1973), 117–124.CrossRefGoogle Scholar