# Journal of the Ramanujan Mathematical Society

Volume 34, Issue 4, December 2019 pp. 427–432.

Existence of Euclidean ideal classes beyond certain rank

**Authors**:
Jyothsnaa Sivaraman

**Author institution:**Institute of Mathematical Sciences, HBNI, C.I.T Campus, Taramani, Chennai 600 113, India

**Summary: **
In his seminal paper on Euclidean ideal classes, Lenstra showed
that under generalised Riemann hypothesis, a number field K has
a Euclidean ideal class if and only if the class group is cyclic.
In [3], the authors show that under certain conditions on
the Hilbert class field of the number field K, for unit rank
greater than or equal to 3, K has a Euclidean ideal class if
and only if the class group is cyclic. The main objective of this
article is to give a short alternate proof of the fact that, under
similar conditions, there exists an integer r ≥ 1 such that
all fields with unit rank greater than or equal to r have a
Euclidean ideal class if and only if the class group is cyclic.
The main novelty of this proof is that we use Brun's sieve as
opposed to the linear sieve as seen traditionally in the context
of this problem.

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