Geodesic
The reduced ideals of a special order in a pure cubic number field
Archivum mathematicum, Tome 56 (2020) no. 3, pp. 171-182
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Let $K=\mathbb{Q}(\theta )$ be a pure cubic field, with $\theta ^3=D$, where $D$ is a cube-free integer. We will determine the reduced ideals of the order $\mathcal{O}=\mathbb{Z}[\theta ]$ of $K$ which coincides with the maximal order of $K$ in the case where $D$ is square-free and $\not\equiv\pm1\pmod9$.
Let $K=\mathbb{Q}(\theta )$ be a pure cubic field, with $\theta ^3=D$, where $D$ is a cube-free integer. We will determine the reduced ideals of the order $\mathcal{O}=\mathbb{Z}[\theta ]$ of $K$ which coincides with the maximal order of $K$ in the case where $D$ is square-free and $\not\equiv\pm1\pmod9$.
DOI : 10.5817/AM2020-3-171
Classification : 11R16, 11R29, 11T71
Keywords: cubic field; reduced ideal 
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Azizi, Abdelmalek; Benamara, Jamal; Ismaili, Moulay Chrif; Talbi, Mohammed. The reduced ideals of a special order in a pure cubic number field. Archivum mathematicum, Tome 56 (2020) no. 3, pp. 171-182. doi: 10.5817/AM2020-3-171

[1] Alaca, S., Williams, K.S.: Introductory algebraic number theory. Cambridge University Press, Cambridge, UK, 2004. | MR

[2] Buchmann, J.A., Scheidler, R., Williams, H.C.: Implementation of a key exchange protocol using real quadratic fields. Advances in Cryptography–EUROCRYPT'90. EUROCRYPT 1990. Lecture Notes in Computer Science (Damgård, I.B., ed.), vol. 473, Springer, Berlin, Heidelberg, 1991, pp. 98–109. | MR

[3] Buchmann, J.A., Williams, H.C.: A key-exchange system based on imaginary quadratic fields. J. Cryptology 1 (1988), 107–118. | DOI | MR

[4] Buchmann, J.A., Williams, H.C.: On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp. 50 (182) (1988), 569–579. | DOI | MR

[5] Buchmann, J.A., Williams, H.C.: A sub exponential algorithm for the determination of class groups and regulators of algebraic number fields. Seminaire de Theorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhauser Boston, Boston, MA, 1990, pp. 27–41. | MR

[6] Cohen, H.: A course in computational algebraic number theory. Springer–Verlag, 1996. | MR

[7] Jacobs, G.T.: Reduced ideals and periodic sequences in pure cubic fields. Ph.D. thesis, University of North Texas, 2015, August 2015, https://digital.library.unt.edu/ark:/67531/metadc804842 | MR

[8] Mollin, R.: Quadratics. CRC Press, Inc., Boca Raton, Florida, 1996. | MR | Zbl

[9] Neukirch, J.: Algebraic Number Theory. Springer–Verlag Berlin, Heidelberg, 1999. | MR | Zbl

[10] Payan, J.: Sur le groupe des classes d’un corps quadratique. Cours de l'institut Fourier 7 (1972), 2–30.

[11] Prabpayak, C.: Orders in pure cubic number fields. Ph.D. thesis, Univ. Graz. Grazer Math. Ber. 361, 2014. | MR

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