PólyaS₃-extensions of ℚ

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.

Relative Pólya group and Pólya dihedral extensions of Q

“…For a number field M , denote the number of ramified primes in M/Q by s M . Leriche [16] In [19], we proved that for a non-Galois cubic field K with Galois closure L, if L is Pólya depending on whether D K > 0, D K < 0 and K pure, or D K < 0 and K non-pure, then s L ≤ 4, s L ≤ 3 or s L ≤ 2, respectively. Also by giving some examples, we showed that these upper bounds are actually sharp, see [19,Section 3].…”

“…The reader is refered to [2,3,4,5,8,9,15,16,17,18,19,24], for some results on Pólya fields and Pólya groups.…”

“…For instance, consider the pure cubic field K = Q( 3 √ 19). One can show that the Galois closure L of K is a Pólya field while Po(K) = Cl(K) ≃ Z/3Z, see[19, Example 2.9]. On the other hand since gcd(h(K), [L :…”

Pólya-Ostrowski Group and Unit Index in Real Biquadratic Fields

“…The main aim of the present paper, is to relate Pólya groups and Hasse unit indices in real biquadratic fields, see Theorem (3.3) below. The reader is refered to [2,3,4,6,6,7,8,9] for some results on Pólya fields and Pólya groups.…”

Ostrowski quotients for finite extensions of number fields

“…Let K = Q( √ −4027)and α be a root of f (x) = x 3 + 10x + 1. One can show that L = K(α) is a cyclic unramified extension of K, see[14, Example 2.14]. By Lemma (4.2), Ost(L/K) nr is trivial andPo(L/K) nr = Po(L/K) = ǫ L/K (Cl(K)).…”