# Cale Bases in Algebraic Orders

• [1] Laboratoire de Mathématiques Pures Université Blaise Pascal Les Cézeaux 63177 AUBIERE CEDEX FRANCE
• Volume: 10, Issue: 1, page 117-131
• ISSN: 1259-1734

top

## Abstract

top
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 $𝒬$ (called a Cale basis) of primary irreducible elements of $R$ such that ${x}^{N}$ has a unique factorization into elements of $𝒬$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}\left(\overline{R}\right)\to \mathrm{Spec}\left(R\right)$ 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

top

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

top
1. D. D. Anderson, K. R. Knopp, R. L. Lewin, Almost Bézout domains II, J. Algebra 167 (1994), 547-556 Zbl0821.13006MR1287059
2. D. D. Anderson, L. A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), 141-154 Zbl0665.13004MR963540
3. D. D. Anderson, M. Zafrullah, Almost Bézout domains, J. Algebra 142 (1991), 285-309 Zbl0749.13013MR1127065
4. S.T. Chapman, F. Halter-Koch, U. Krause, Inside factorial monoids and integral domains, J. Algebra 252 (2002), 350-375 Zbl1087.13510MR1925142
5. T. Dumitrescu, Y. Lequain, J. L. Mott, M. Zafrullah, Almost GCD domains of finite $t$-character, J. Algebra 245 (2001), 161-181 Zbl1094.13537MR1868187
6. H. M. Edwards, Fermat’s last Theorem, (1977), Springer GTM, Berlin Zbl0355.12001MR616635
7. A. Faisant, Interprétation factorielle du nombre de classes dans les ordres des corps quadratiques, Ann. Math. Blaise Pascal 7 (2) (2000), 13-18 Zbl1013.11071MR1815164
8. A. Geroldinger, F. Halter-Koch, J. Kaczorowski, Non-unique factorizations in orders of global fields, J. Reine Angew. Math. 459 (1995), 89-118 Zbl0812.11061MR1319518
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
11. M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29-62 Zbl0587.13010MR788672
12. P. Zanardo, U. Zannier, The class semigroup of orders in number fields, Math. Proc. Cambridge Philos. Soc. 115 (1994), 379-391 Zbl0828.11068MR1269926

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.