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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  • Y.-F. Bilu, J.-M. Deshouillers, S. Gun and F. Luca, Random orderings in modulus of consecutive Hecke eigenvalues of primitive forms, to appear in Compositio math.

  • D. A. Clark and M. R. Murty, The Euclidean algorithm for Galois extensions of Q, J. Reine Angew. Math., 459 (1995) 151–162.

  • J.-M. Deshouillers, S. Gun and J. Sivaraman, On Euclidean Ideal classes in certain Abelian extensions, submitted.

  • H. Graves and M. R. Murty, A family of number fields with unit rank at least 4 that has Euclidean ideals, Proc. Amer. Math. Soc., 141 (2013) 2979–2990.

  • R. Gupta, M. R. Murty and V. K. Murty, The Euclidean algorithm for S-integers, Number Theory (Montreal 1985), CMS Conf. Proc. 7, American Mathematical Society, Providence (1987) 189–201.

  • H. Graves, Growth results and Euclidean ideals, J. Number Theory, 133 no. 8, (2013) 2756–2769.

  • H. W. Lenstra, Jr., Euclidean ideal classes, Ast´erisque, 61 (1979) 121–131.

  • M. Harper and M. R. Murty, Euclidean rings of algebraic integers, Canad. J. Math., 56 no. 1, (2004) 71–76.

  • M. Harper, Z[√14] is Euclidean, Canad. J. Math., 56 no. 1, (2004) 55–70.

  • M. R. Murty and K. L. Petersen, The Euclidean algorithm for number fields and primitive ischolar_mains, Proc. Amer. Math. Soc., 141 (2013) 181–190.

  • W. Narkiewicz, Euclidean algorithm in small abelian fields, Funct. Approx. Comment. Math., 37 no. 2, (2007) 337–340.

  • P. J. Weinberger, On Euclidean rings of algebraic integers, Proc. Symposia Pure Math., 24 (1972) 321–332.


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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