Non-norm-Euclidean fields in basic \(Z_{l}\)-extensions. (English) Zbl 1415.11153

Summary: 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.


11R20 Other abelian and metabelian extensions
11R29 Class numbers, class groups, discriminants
Full Text: DOI Euclid


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