On the Pólya Group of Some Imaginary Biquadratic Fields

Taous, Mohammed

doi:10.1007/978-3-319-32902-4_12

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…The study of Pólya fields has now become very important in the theory of integer valued polynomials. Some interesting results can be seen in [63], [67], [68], [70], [69], [107], [108] and [126]. A sufficient condition for a number field to be a Pólya field can be obtained from FD sequences and fixed divisors as follows.…”

Section: Fixed Divisor Sequences and Related Notionsmentioning

confidence: 99%

A Survey on Fixed Divisors

Prasad¹,

Rajkumar²,

Reddy³

2019

Confluentes Mathematici

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In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n × n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials NotationsWe fix the notations for the whole paper. R = Integral Domain K = Field of fractions of R N (I) = Cardinality of R/I (Norm of an ideal I ⊆ R) W = {0, 1, 2, 3, . . .} A[x] = Ring of polynomials in n variables (= A[x 1 , . . . , x n ]) with coefficients in the ring A S = Arbitrary (or given) subset of R n such that no non-zero polynomial in K[x] maps it to zero S = S in case when n = 1 Int(S, R) = Polynomials in K[x] mapping S back to R ν k (S) = Bhargava's (generalized) factorial of index k k! S = k th generalized factorial in several variables M m (S) = Set of all m × m matrices with entries in S p = positive prime number Z p = p-adic integers ord p (n) = p-adic ordinal (valuation) of n ∈ Z.

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Section: Fixed Divisor Sequences and Related Notionsmentioning

confidence: 99%

A Survey on Fixed Divisors

Prasad¹,

Rajkumar²,

Reddy³

2019

Confluentes Mathematici

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“…In Section 4 we recall some classical results about the order of this group of strongly ambiguous classes. About Pólya groups we may also cite more recent papers: [2], [5], [9], [26], [27], [30].…”

Section: Pólya Groupsmentioning

confidence: 99%

From Polya fields to Polya groups: (I) Galoisian extensions

Chabert

2018

Preprint

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The Pólya group of a number field K is the subgroup of the class group of K generated by the classes of the products of the maximal ideals with same norm. A Pólya field is a number field whose Pólya group is trivial. Our purpose is to start with known assertions about Pólya fields to find results concerning Pólya groups. This first paper inspired by Zantema's results focuses on the Pólya group of the compositum of two galoisian extensions of Q.

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On dihedral Pólya fields

Tougma

2025

Journal of Number Theory

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On the Pólya Group of Some Imaginary Biquadratic Fields

Cited by 3 publications

References 9 publications

A Survey on Fixed Divisors

A Survey on Fixed Divisors

From Polya fields to Polya groups: (I) Galoisian extensions

On dihedral Pólya fields

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