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Enumeration of quartic fields of small discriminant. (English) Zbl 0788.11060
Summary: We have previously enumerated totally real (signature 4) [J. Buchmann and D. Ford, Math. Comput. 52, 161-174 (1989; Zbl 0668.12001)] and totally complex (signature 0) [D. Ford, Computational Number Theory, Debrecen 1989, 129-138 (1991; Zbl 0729.11051)] fields.
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.

MSC:
11Y40 Algebraic number theory computations
11-04 Software, source code, etc. for problems pertaining to number theory
11R16 Cubic and quartic extensions
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[1] Johannes Buchmann and David Ford, On the computation of totally real quartic fields of small discriminant, Math. Comp. 52 (1989), no. 185, 161 – 174. · Zbl 0668.12001
[2] D. Ford, Enumeration of totally complex quartic fields of small discriminant, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 129 – 138. · Zbl 0729.11051
[3] Irving Gerst and John Brillhart, On the prime divisors of polynomials, Amer. Math. Monthly 78 (1971), 250 – 266. · Zbl 0214.30604
[4] H. J. Godwin, On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2) 8 (1957), 214 – 222. · Zbl 0079.05704
[5] H. J. Godwin, On quartic fields of signature one with small discriminant. II, Math. Comp. 42 (1984), no. 166, 707 – 711. , https://doi.org/10.1090/S0025-5718-1984-0736462-9 H. J. Godwin, Corrigenda: ”On quartic fields of signature one with small discriminant. II”, Math. Comp. 43 (1984), no. 168, 621. · Zbl 0535.12003
[6] A. K. Lenstra, Factorization of polynomials, Computational methods in number theory, Part I, Math. Centre Tracts, vol. 154, Math. Centrum, Amsterdam, 1982, pp. 169 – 198.
[7] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515 – 534. · Zbl 0488.12001
[8] Michael Pohst, On the computation of number fields of small discriminants including the minimum discriminants of sixth degree fields, J. Number Theory 14 (1982), no. 1, 99 – 117. · Zbl 0478.12005
[9] M. Pohst, On computing isomorphisms of equation orders, Math. Comp. 48 (1987), no. 177, 309 – 314. · Zbl 0632.12001
[10] I. Schur, Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, S.-B. Berlin Math. Ges. 11 (1912), 40-50. · JFM 43.0261.02
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