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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 28 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1451330563

Digital Object Identifier
doi:10.3792/pjaa.92.23

Mathematical Reviews number (MathSciNet)
MR3447746

Zentralblatt MATH identifier
06586131

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

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

Citation

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. https://projecteuclid.org/euclid.pja/1451330563


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