Proceedings of the Japan Academy, Series A, Mathematical Sciences

Non-norm-Euclidean fields in basic $\mathbf{Z}_{l}$-extensions

Kuniaki Horie and Mitsuko Horie

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We shall deal with infinite towers of cyclic fields of genus number 1 in which a prime number $l\geq 5$ is totally ramified. Our main result states that, if $m$ is a positive divisor of $l-1$ less than $(l-1)/2$, then for any positive integer $n$, the cyclic field of degree $ml^{n}$ with conductor $l^{n+1}$ is not norm-Euclidean. In particular, it follows that, for any positive integer $n$, the (real) cyclic field of degree $l^{n}$ with conductor $l^{n+1}$ is not norm-Euclidean and that the (imaginary) cyclic field of degree 14 with conductor 49, whose class number is known to equal 1, is not norm-Euclidean.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 92, Number 1 (2016), 23-26.

First available in Project Euclid: 28 December 2015

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Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Secondary: 11R20: Other abelian and metabelian extensions 11R29: Class numbers, class groups, discriminants

Norm-Euclidean field cyclic field class number genus number basic $\mathbf{Z}_{l}$-extension


Horie, Kuniaki; Horie, Mitsuko. Non-norm-Euclidean fields in basic $\mathbf{Z}_{l}$-extensions. Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 1, 23--26. doi:10.3792/pjaa.92.23.

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  • H. Bauer, Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper, J. Number Theory 1 (1969), 161–162.
  • J. Buhler, C. Pomerance and L. Robertson, Heuristics for class numbers of prime-power real cyclotomic fields, in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, Fields Inst. Commun., 41, Amer. Math. Soc., Providence, RI, 2004, pp. 149–157.
  • J.-P. Cerri, De l'euclidianité de $\mathbf{Q}(\sqrt{\smash{2+\sqrt{2+\sqrt{2}}}\mathstrut})$ et $\mathbf{Q}(\sqrt{2+\sqrt{2}})$ pour la norme, J. Théor. Nombres Bordeaux 12 (2000), no. 1, 103–126.
  • H. Cohn and J. Deutsch, Use of a computer scan to prove $\mathbf{Q}(\sqrt{2+\sqrt{2}})$ and $\mathbf{Q}(\sqrt{3+\sqrt{2}})$ are Euclidean, Math. Comp. 46 (1986), no. 173, 295–299.
  • H. Davenport, On the product of three non-homogeneous linear forms, Math. Proc. Cambridge Philos. Soc. 43 (1947), 137–152.
  • Y. Furuta, The genus field and genus number in algebraic number fields, Nagoya Math. J. 29 (1967), 281–285.
  • H. Heilbronn, On Euclid's algorithm in cyclic fields, Canadian J. Math. 3 (1951), 257–268.
  • K. Horie and M. Horie, The $l$-class group of the $\mathbf{Z}_{p}$-extension over the rational field, J. Math. Soc. Japan 64 (2012), no. 4, 1071–1089.
  • S. Iyanaga and T. Tamagawa, Sur la théorie du corps de classes sur le corps des nombres rationnels, J. Math. Soc. Japan 3 (1951), 220–227.
  • H. W. Leopoldt, Zur Geschlechtertheorie in abelschen Zahlkörpern, Math. Nachr. 9 (1953), 351–362.
  • F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693–707.
  • J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio Math. 37 (1978), no. 3, 297–319.
  • K. J. McGown, Norm-Euclidean cyclic fields of prime degree, Int. J. Number Theory 8 (2012), no. 1, 227–254.
  • J. C. Miller, Class numbers in cyclotomic $\mathbf{Z}_{p}$-extensions, J. Number Theory 150 (2015), 47–73.
  • K. Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899–921.