Abstract
We give here the complete table of octic fields with a quartic subfield with minimum discriminants for all signatures and for all the possible Galois groups. Moreover we give some pairs of octic fields arithmetically equivalent (i.e. with the same Dedekind zeta function).
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References
G. Butler and J. McKay: The transitive groups of degree up to eleven. Comm. in Algebra 11 (1983) 863–911
H. Cohen, F. Diaz y Diaz and M. Olivier: Algorithmic methods for finitely generated Abelian groups. Submitted to J. of Symb. Comp. (1997)
H. Cohen, F. Diaz y Diaz and M. Olivier: Tables of Octic Fields with a Quartic Subfield. Submitted to Math. of Comp. (1998)
J. Conway, A. Hulpke and J. McKay: Names and generators for the transitive groups of degree up to 15. Preprint (1996)
F. Diaz y Diaz: Petits discriminants des corps de nombres totalement imaginaires de degré 8. J. of Number Th. 25 (1987) 34–52
Y. Eichenlaub: Problèmes effectifs de théorie de Galois en degrés 8 à 11. Thèse Université Bordeaux I (1996)
Y. Eichenlaub and M. Olivier: Computation of Galois groups for polynomials with degree up to eleven. Submitted to Math. of Comp. (1997)
F. Gassmann: Bemerkungen zur Vorstehenden Arbeit von Hurwitz. Mat. Z. 25 (1926) 665–675
A. Jehanne: Sur les extensions de Q à groupe de Galois S 4 et S 4. Acta Arith. LXIX (1995) 259–276
S.-H. Kwon: Sur les discriminants minimaux des corps quaternioniens. Arch. Math. 67 (1996) 119–125
R. Perlis: On the equation ζK(s) = ζk′(s). J. Number Th. 9 (1977) 342–360
M. Pohst, J. Martinet and F. Diaz y Diaz: The Minimum Discriminant of Totally Real Octic Fields. J. Number Th. 36 (1990) 145–159
B. de Smit and R. Perlis: Zeta functions do not determine class numbers. Bull. Am. Math. Soc. 31 (1994) 213–215
G. W. Smith: Some polynomials over Q(t) and their Galois groups. Preprint (1993)
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© 1998 Springer-Verlag Berlin Heidelberg
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Cohen, H., Diaz y Diaz, F., Olivier, M. (1998). Imprimitive octic fields with small discriminants. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054876
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DOI: https://doi.org/10.1007/BFb0054876
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