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Computing Square Roots in Quaternion Algebras

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Języki publikacji
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Abstrakty
EN
We present an explicit algorithmic method for computing square roots in quaternion algebras over global fields of characteristic different from 2.
Wydawca
Rocznik
Strony
1--15
Opis fizyczny
Bibliogr. 17 poz.
Twórcy
  • Institute of Mathematics University of Silesia in Katowice ul. Bankowa 14, 40-007 Katowice, Poland
Bibliografia
  • [1] Heath T. A history of Greek mathematics. Vol. I. Dover Publications, Inc., New York, 1981. ISBN 0-486-24073-8. From Thales to Euclid, Corrected reprint of the 1921 original. [2] Niven I. The roots of a quaternion. Amer. Math. Monthly, 1942. 49:386-388. doi:10.2307/2303134. URL https://doi.org/10.2307/2303134.
  • [3] Vignéras MF. Arithm´etique des algebra’s de quaternions, volume 800 of Lecture Notes in Mathematics. Springer, Berlin, 1980. ISBN 3-540-09983-2.
  • [4] Voight J. Quaternion algebras, volume 288 of Graduate Texts in Mathematics. Springer, Cham, [2021]©2021. ISBN 978-3-030-56692-0; 978-3-030-56694-4. doi:10.1007/978-3-030-56694-4. URL https: //doi.org/10.1007/978-3-030-56694-4.
  • [5] Lam TY. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-1095-2.
  • [6] Gordon B, Motzkin TS. On the zeros of polynomials over division rings. Trans. Amer. Math. Soc., 1965. 116:218-226. doi:10.2307/1994114. URL https://doi.org/10.2307/1994114.
  • [7] Cohen H. Advanced topics in computational number theory, volume 193 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. ISBN 0-387-98727-4. doi:10.1007/978-1-4419-8489-0. URL https://doi.org/10.1007/978-1-4419-8489-0.
  • [8] Fieker C, Jurk A, Pohst M. On solving relative norm equations in algebraic number fields. Math. Comp., 1997. 66(217):399-410. doi:10.1090/S0025-5718-97-00761-8. URL https://doi.org/10.1090/S0025-5718-97-00761-8.
  • [9] Fincke U, Pohst M. A procedure for determining algebraic integers of given norm. In: Computer algebra (London, 1983), volume 162 of Lecture Notes in Comput. Sci., pp. 194-202. Springer, Berlin, 1983. doi: 10.1007/3-540-12868-9\ 103. URL https://doi.org/10.1007/3-540-12868-9_103.
  • [10] Garbanati DA. An algorithm for finding an algebraic number whose norm is a given rational number. J. Reine Angew. Math., 1980. 316:1-13. doi:10.1515/crll.1980.316.1. URL https://doi.org/10.1515/crll.1980.316.1.
  • [11] Simon D. Solving norm equations in relative number fields using S-units. Math. Comp., 2002. 71(239):1287-1305. doi:10.1090/S0025-5718-02-01309-1. URL https://doi.org/10.1090/S0025-5718-02-01309-1.
  • [12] Czogała A. Witt rings of Hasse domains of global fields. J. Algebra, 2001. 244(2):604-630. doi: 10.1006/jabr.2001.8918. URL https://doi.org/10.1006/jabr.2001.8918.
  • [13] Cannon J, Bosma W, Fieker C, (eds) AS. Handbook of Magma Functions, 2.26-4 edition, 2021.
  • [14] Koprowski P. Computing singular elements modulo squares. Fund. Inform., 2021. 179(3):227-238. doi:10.3233/fi-2021-2022. URL https://doi.org/10.3233/fi-2021-2022.
  • [15] Koprowski P, Rothkegel B. The anisotropic part of a quadratic form over a number field. J. Symbolic Comput., 2023. 115:39-52. doi:10.1016/j.jsc.2022.07.003. URL https://doi.org/10.1016/j.jsc.2022.07.003.
  • [16] Koprowski P, Czogała A. Computing with quadratic forms over number fields. J. Symbolic Comput., 2018. 89:129-145. doi:10.1016/j.jsc.2017.11.009. URL https://doi.org/10.1016/j.jsc.2017.11.009.
  • [17] Leep D, Wadsworth A. The Hasse norm theorem mod squares. J. Number Theory, 1992. 42(3):337-348. doi:10.1016/0022-314X(92)90098-A. URL https://doi.org/10.1016/0022-314X(92)90098-A.
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