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

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.