When is R[θ] integrally closed?

Khanduja, Sudesh K.; Jhorar, Bablesh

doi:10.1142/s0219498816500912

Cited by 10 publications

(7 citation statements)

References 4 publications

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“…Precisely, we prove: The following result proved in [7,Theorem 1.3] can be quickly deduced from Corollary 1.2.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 77%

On power basis of a class of number fields

Jakhar¹,

Kaur²,

Kumar³

2023

Preprint

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Let f (x) = x n + ax 2 + bx + c ∈ Z[x] be an irreducible polynomial with b 2 = 4ac and let K = Q(θ) be an algebraic number field defined by a complex root θ of f (x). Let Z K deonote the ring of algebraic integers of K. The aim of this paper is to provide the necessary and sufficient conditions involving only a, c and n for a given prime p to divide the index of the subgroup Z[θ] in Z K . As a consequence, we provide families of monogenic algebraic number fields. Further, when2010 Mathematics Subject Classification. 11R04; 11R29. Key words and phrases. Ring of algebraic integers; Integral basis and discriminant; Monogenic number fields.By virtue of Corollary 1.2, Z K = Z[θ] if and only if for each prime p dividing D f , p 2 does not divide c when p | c and if p ∤ c, then p 2 ∤ D f . For example, if we take n = 5 and c = ±1 is a squarefree integer, then {1, θ, θ 2 , θ 3 , θ 4 } is an integral basis of K if and only if (3125 − 108c) is squarefree. It can be verified that (3125 − 108c) is squarefree when c = −3, 5, 13, 17, 21.

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“…Precisely, we prove: The following result proved in [7,Theorem 1.3] can be quickly deduced from Corollary 1.2.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 77%

On power basis of a class of number fields

Jakhar¹,

Kaur²,

Kumar³

2023

Preprint

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“…In the case of rings of integers of number fields, the following theorem strongly enhances THEOREM 1.3. Besides, THEOREM 1.4 generalizes the relevant results in [7] and [12]. For a ring of integers R, by ν p (s) we mean ν p (sR) for s ∈ R and a nonzero prime ideal p of R. THEOREM 1.4.…”

Section: Theorem 11 With the Above Assumptions And Notationsmentioning

confidence: 73%

“…Ershov, in [5], gave yet a generalization of this criterion to extensions of rings of valuation. This criterion had, and still have, important applications in many relevant areas such as (but not limited to) the study of prime ideal factorizations in Dedekind rings, the computation of discriminants of number fields, and the existence of integral power bases in extensions of Dedekind rings (see for instance [1], [7], [11], and [12]).…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

“…If we let R = Z and K = Q in THEOREM 1.4, then [7,Thoerem 1.3] can be phrased as follows: Z[α] is integrally closed if and only if, for every rational prime p, either ν p (u) = 0 or ν p (u) = 1 and ν p (u p νp(n) − u) = 1. The following corollary is an improvement of [7,Theorem 1.3]. COROLLARY 1.5.…”

Section: Theorem 11 With the Above Assumptions And Notationsmentioning

confidence: 99%

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On the Integral Closedness of $R[α]$

Deajim,

Fadil

2018

Preprint

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Let R be a Dedekind ring, K its quotient field, and L = K(α) a finite field extension of K defined by a monic irreducible polynomial f (x) ∈ R[x]. We give an easy version of Dedekind's criterion which computationally improves those versions know in the literature. We further use this result to give a sufficient condition for the integral closedness of R[α] when f (x) = x n − a. In case R is a ring of integers of a number field, we give yet sufficient and necessary conditions for this to hold, generalizing and improving in both cases some known results in this direction. Some highlighting examples are also given.

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“…A similar result holds with applications in more general rings, namely P rü f e r domains (cf. [9]). In This work, we are interested in another way, namely computation of integral bases.…”

Section: Resultsmentioning

confidence: 99%

Dedekind’s Criterion and Integral Bases

Fadil

2019

Tatra Mountains Mathematical Publications

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Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤL be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤL: R[α]], for every prime p of R. Some computational examples are given for R = ℤ.

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When is R[θ] integrally closed?

Cited by 10 publications

References 4 publications

On power basis of a class of number fields

On power basis of a class of number fields

On the Integral Closedness of $R[α]$

Dedekind’s Criterion and Integral Bases

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