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Totally real bi-quadratic fields with large Pólya groups

Abstract

For an algebraic number field K with ring of integers \(\mathcal {O}_{K}\), an important subgroup of the ideal class group \(Cl_{K}\) is the Pólya group, denoted by Po(K), which measures the failure of the \(\mathcal {O}_{K}\)-module \(Int(\mathcal {O}_{K})\) of integer-valued polynomials on \(\mathcal {O}_{K}\) from admitting a regular basis. In this paper, we prove that for any integer \(n \ge 2\), there are infinitely many totally real bi-quadratic fields K with \(Po(K) \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{n}\). In fact, we explicitly construct such an infinite family of number fields. This also provides an infinite family of bi-quadratic fields with ideal class groups of 2-ranks at least n.

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Acknowledgements

We gratefully thank the anonymous referee for a meticulous reading of the article and providing a detailed report on the earlier version. This improved the quality and the readability of the article to a great extent. It is a pleasure for the first author to thank Indian Institute of Technology Guwahati for providing necessary facilities to carry out this research work. He gratefully acknowledges the National Board of Higher Mathematics for providing financial support (Order No. 0204/16(12)/2020/R & D-II/10925). The second author would like to thank MATRICS, SERB for their research grant (Grant No. MTR/2020/000467).

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Correspondence to Jaitra Chattopadhyay.

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Chattopadhyay, J., Saikia, A. Totally real bi-quadratic fields with large Pólya groups. Res. number theory 8, 30 (2022). https://doi.org/10.1007/s40993-022-00327-8

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  • DOI: https://doi.org/10.1007/s40993-022-00327-8

Keywords

  • Pólya field
  • Pólya group
  • Real bi-quadratic field
  • Galois cohomology

Keywords

  • Pólya field
  • Pólya group
  • Real bi-quadratic field
  • Galois cohomology

2020 Mathematics Subject Classification:

  • Primary 11R29
  • 11R34
  • Secondary 13F20