Journal of the Ramanujan Mathematical Society

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Existence of Euclidean Ideal Classes Beyond Certain Rank


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1 Institute of Mathematical Sciences, HBNI, C.I.T Campus, Taramani, Chennai 600 113, India
     

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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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  • Existence of Euclidean Ideal Classes Beyond Certain Rank

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Authors

Jyothsnaa Sivaraman
Institute of Mathematical Sciences, HBNI, C.I.T Campus, Taramani, Chennai 600 113, India

Abstract


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.

References