Cale Bases in Algebraic Orders

Martine Picavet-L’Hermitte[1]

  • [1] Laboratoire de Mathématiques Pures Université Blaise Pascal Les Cézeaux 63177 AUBIERE CEDEX FRANCE

Annales mathématiques Blaise Pascal (2003)

  • Volume: 10, Issue: 1, page 117-131
  • ISSN: 1259-1734

Abstract

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Let R be a non-maximal order in a finite algebraic number field with integral closure R ¯ . Although R is not a unique factorization domain, we obtain a positive integer N and a family 𝒬 (called a Cale basis) of primary irreducible elements of R such that x N has a unique factorization into elements of 𝒬 for each x ∈ R coprime with the conductor of R . Moreover, this property holds for each nonzero x ∈ R when the natural map Spec ( R ¯ ) → Spec ( R ) is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.

How to cite

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Picavet-L’Hermitte, Martine. "Cale Bases in Algebraic Orders." Annales mathématiques Blaise Pascal 10.1 (2003): 117-131. <http://eudml.org/doc/10480>.

@article{Picavet2003,
abstract = {Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline\{R\}$. Although $R$ is not a unique factorization domain, we obtain a positive integer $N$ and a family $\{\mathcal\{Q\}\}$ (called a Cale basis) of primary irreducible elements of $R$ such that $x^N$ has a unique factorization into elements of $\{\mathcal\{Q\}\}$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm\{Spec\}(\overline\{R\})\rightarrow \mathrm\{Spec\}(R)$ is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.},
affiliation = {Laboratoire de Mathématiques Pures Université Blaise Pascal Les Cézeaux 63177 AUBIERE CEDEX FRANCE},
author = {Picavet-L’Hermitte, Martine},
journal = {Annales mathématiques Blaise Pascal},
keywords = {orders; Cale bases; factorization},
language = {eng},
month = {1},
number = {1},
pages = {117-131},
publisher = {Annales mathématiques Blaise Pascal},
title = {Cale Bases in Algebraic Orders},
url = {http://eudml.org/doc/10480},
volume = {10},
year = {2003},
}

TY - JOUR
AU - Picavet-L’Hermitte, Martine
TI - Cale Bases in Algebraic Orders
JO - Annales mathématiques Blaise Pascal
DA - 2003/1//
PB - Annales mathématiques Blaise Pascal
VL - 10
IS - 1
SP - 117
EP - 131
AB - Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline{R}$. Although $R$ is not a unique factorization domain, we obtain a positive integer $N$ and a family ${\mathcal{Q}}$ (called a Cale basis) of primary irreducible elements of $R$ such that $x^N$ has a unique factorization into elements of ${\mathcal{Q}}$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}(\overline{R})\rightarrow \mathrm{Spec}(R)$ is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.
LA - eng
KW - orders; Cale bases; factorization
UR - http://eudml.org/doc/10480
ER -

References

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  9. M. Picavet-L’Hermitte, Factorization in some orders with a PID as integral closure, Algebraic Number Theory and Diophantine Analysis (2000), 365-390, Halter-KochF.F. Zbl0971.13016MR1770474
  10. M. Picavet-L’Hermitte, Weakly factorial quadratic orders, Arab. J. Sci. and Engineering 26 (2001), 171-186 Zbl1271.13003MR1843467
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