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Keywords:
cubic field; reduced ideal
cubic field; reduced ideal
Summary:
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 $\not\equiv\pm1\pmod9$.
References:
[1] Alaca, S., Williams, K.S.: Introductory algebraic number theory. Cambridge University Press, Cambridge, UK, 2004. MR 2031707
[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 1102474
[3] Buchmann, J.A., Williams, H.C.: A key-exchange system based on imaginary quadratic fields. J. Cryptology 1 (1988), 107–118. DOI 10.1007/BF02351719 | MR 0972575
[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 10.1090/S0025-5718-1988-0929554-6 | MR 0929554
[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 1104698
[6] Cohen, H.: A course in computational algebraic number theory. Springer–Verlag, 1996. MR 1228206
[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 3503469
[8] Mollin, R.: Quadratics. CRC Press, Inc., Boca Raton, Florida, 1996. MR 1383823 | Zbl 0888.11041
[9] Neukirch, J.: Algebraic Number Theory. Springer–Verlag Berlin, Heidelberg, 1999. MR 1697859 | Zbl 0956.11021
[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 3364308
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