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Pólya S3-extensions of ℚ

Part of: Algebraic number theory: global fields Arithmetic rings and other special rings

Published online by Cambridge University Press:  27 December 2018

Abbas Maarefparvar  and
Ali Rajaei
Abbas Maarefparvar
Affiliation:
Department of Mathematics, Tarbiat Modares University, Tehran 14115-134, Iran (a.maarefparvar@modares.ac.ir; alirajaei@modares.ac.ir)
Ali Rajaei
Affiliation:
Department of Mathematics, Tarbiat Modares University, Tehran 14115-134, Iran (a.maarefparvar@modares.ac.ir; alirajaei@modares.ac.ir)
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Abstract

A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

Keywords

integer-valued polynomialsPólya fieldsnon-Galois cubic fieldsS3-fieldsS3-extensions

MSC classification

Primary: 11R04: Algebraic numbers; rings of algebraic integers
Secondary: 11R16: Cubic and quartic extensions 11R29: Class numbers, class groups, discriminants 11R37: Class field theory 11R34: Galois cohomology 13F20: Polynomial rings and ideals; rings of integer-valued polynomials
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1421 - 1433
Copyright
Copyright © Royal Society of Edinburgh 2018 

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