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Pólya S3-extensions of ℚ

Published online by Cambridge University Press:  27 December 2018

Abbas Maarefparvar
Affiliation:
Department of Mathematics, Tarbiat Modares University, Tehran 14115-134, Iran ([email protected]; [email protected])
Ali Rajaei
Affiliation:
Department of Mathematics, Tarbiat Modares University, Tehran 14115-134, Iran ([email protected]; [email protected])

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.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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