Translator Disclaimer
January 2016 Non-norm-Euclidean fields in basic $\mathbf{Z}_{l}$-extensions
Kuniaki Horie, Mitsuko Horie
Proc. Japan Acad. Ser. A Math. Sci. 92(1): 23-26 (January 2016). DOI: 10.3792/pjaa.92.23

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.

Citation

Download Citation

Kuniaki Horie. Mitsuko Horie. "Non-norm-Euclidean fields in basic $\mathbf{Z}_{l}$-extensions." Proc. Japan Acad. Ser. A Math. Sci. 92 (1) 23 - 26, January 2016. https://doi.org/10.3792/pjaa.92.23

Information

Published: January 2016
First available in Project Euclid: 28 December 2015

zbMATH: 06586131
MathSciNet: MR3447746
Digital Object Identifier: 10.3792/pjaa.92.23

Subjects:
Primary: 11A05
Secondary: 11R20, 11R29

Rights: Copyright © 2016 The Japan Academy

JOURNAL ARTICLE
4 PAGES


SHARE
Vol.92 • No. 1 • January 2016
Back to Top