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Enumeration of quartic fields of small discriminant

Authors: Johannes Buchmann, David Ford and Michael Pohst
Journal: Math. Comp. 61 (1993), 873-879
MSC: Primary 11R16; Secondary 11R32, 11R80
MathSciNet review: 1176706
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Abstract: With the mixed-type case now completed, all algebraic number fields of degree 4 with absolute discriminant $ < {10^6}$ have been enumerated. Methods from the totally real and totally complex cases were used without major modification. Isomorphism of fields was determined by a method similar to one of Lenstra. The $ {T_2}$ criterion of Pohst was applied to reduce the number of redundant examples.

References [Enhancements On Off] (What's this?)

  • [1] J. Buchmann and D. Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), 161-174. MR 946599 (89f:11147)
  • [2] D. Ford, Enumeration of totally complex quartic fields of small discriminant, Computational Number Theory, Proceedings of the Colloquium on Computational Number Theory, Debrecen (Hungary), 1989 (A. Pethö, M. Pohst, H. C. Williams, and H. G. Zimmer, eds.), de Gruyter, Berlin and New York, 1991, pp. 129-138. MR 1151860 (93b:11140)
  • [3] I. Gerst and J. Brillhart, On the prime divisors of a polynomial, Amer. Math. Monthly 78 (1971), 250-266. MR 0279071 (43:4797)
  • [4] H. J. Godwin, On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2) 8 (1957), 214-222. MR 0097375 (20:3844)
  • [5] -, On quartic fields of signature one with small discriminant. II, Math. Comp. 42 (1984), 707-711. MR 736462 (85i:11092a)
  • [6] A. K. Lenstra, Lattices and factorization of polynomials over algebraic number fields, Proceedings Eurocam 82, Lecture Notes in Comput. Sci., vol. 144, Springer, Berlin, 1982, pp. 32-39. MR 700263 (85a:12002)
  • [7] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515-534. MR 682664 (84a:12002)
  • [8] M. Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), 99-117. MR 644904 (83g:12009)
  • [9] -, On computing isomorphisms of equation orders, Math. Comp. 48 (1987), 309-314. MR 866116 (88b:11066)
  • [10] I. Schur, Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, S.-B. Berlin Math. Ges. 11 (1912), 40-50.

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Article copyright: © Copyright 1993 American Mathematical Society

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