|
Tables of octic fields with a quartic subfield
Authors:
H. Cohen, F. Diaz y Diaz and M. Olivier
Journal:
Math. Comp. 68 (1999), 1701-1716
MSC (1991):
Primary 11R37, 11Y40
Posted:
February 24, 1999
MathSciNet review:
1642813
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.
- [Be]
K.
Belabas, A fast algorithm to compute cubic
fields, Math. Comp. 66
(1997), no. 219, 1213–1237.
MR
1415795 (97m:11159), http://dx.doi.org/10.1090/S0025-5718-97-00846-6
- [Bu-Mc]
Gregory
Butler and John
McKay, The transitive groups of degree up to eleven, Comm.
Algebra 11 (1983), no. 8, 863–911. MR 695893
(84f:20005), http://dx.doi.org/10.1080/00927878308822884
- [Co]
Henri
Cohen, A course in computational algebraic number theory,
Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin,
1993. MR
1228206 (94i:11105)
- [Co-Di]
Henri
Cohen and Francisco
Diaz y Diaz, A polynomial reduction algorithm, Sém.
Théor. Nombres Bordeaux (2) 3 (1991), no. 2,
351–360 (English, with French summary). MR 1149802
(93a:11107)
- [Co-Di-Ol]
H.
Cohen, F.
Diaz y Diaz, and M.
Olivier, Computing ray class groups, conductors
and discriminants, Math. Comp.
67 (1998), no. 222, 773–795. MR 1443117
(98g:11128), http://dx.doi.org/10.1090/S0025-5718-98-00912-0
- [Co-Hu-Mc]
J. Conway, A. Hulpke and J. McKay, On transitive permutation groups, LMS J. Comput. Math., 1 (1998), 1-8.
- [Di-Ol]
F. Diaz y Diaz and M. Olivier, Imprimitive ninth-degree number fields with small discriminants, Math. Comp. 64 (1995), 305-321. MR 95e:11153
- [Ei]
Y. Eichenlaub, Problèmes effectifs de théorie de Galois en degrés
 à  , Thèse, Université Bordeaux I, 1996.
- [Ei-Ol]
Y. Eichenlaub and M. Olivier, Computation of Galois groups for polynomials with degree up to eleven (1996), Preprint.
- [Ga]
F. Gassmann, Bemerkungen zur Vorstehenden Arbeit von Hurwitz, Math. Z. 25 (1926), 665-675.
- [Ha]
Helmut
Hasse, Bericht über neuere Untersuchungen und Probleme aus der
Theorie der algebraischen Zahlkörper. Teil I:
Klassenkörpertheorie. Teil Ia: Beweise zu Teil I. Teil II:
Reziprozitätsgesetz, Dritte Auflage, Physica-Verlag,
Würzburg, 1970 (German). MR 0266893
(42 #1795)
- [Je]
Kevin
Hutchinson, Norm groups of global fields, J. Number Theory
50 (1995), no. 2, 323–328. MR 1316827
(95m:11126), http://dx.doi.org/10.1006/jnth.1995.1026
- [Kw]
S.-H.
Kwon, Sur les discriminants minimaux des corps quaternioniens,
Arch. Math. (Basel) 67 (1996), no. 2, 119–125
(French). MR
1399827 (97d:11165), http://dx.doi.org/10.1007/BF01268925
- [Pe]
Robert
Perlis, On the equation
𝜁_{𝐾}(𝑠)=𝜁_{𝐾’}(𝑠),
J. Number Theory 9 (1977), no. 3, 342–360. MR 0447188
(56 #5503)
- [Po-Za]
M.
Pohst and H.
Zassenhaus, Algorithmic algebraic number theory, Encyclopedia
of Mathematics and its Applications, vol. 30, Cambridge University
Press, Cambridge, 1989. MR 1033013
(92b:11074)
- [Ro]
Xavier-François
Roblot, Unités de Stark et corps de classes de Hilbert,
C. R. Acad. Sci. Paris Sér. I Math. 323 (1996),
no. 11, 1165–1168 (French, with English and French summaries).
MR
1423444 (97k:11162)
- [Sm-Pe]
Bart
de Smit and Robert
Perlis, Zeta functions do not determine class
numbers, Bull. Amer. Math. Soc. (N.S.)
31 (1994), no. 2,
213–215. MR 1260520
(95a:11100), http://dx.doi.org/10.1090/S0273-0979-1994-00520-8
- [Sm]
G. W. Smith, Some polynomials over
 and their Galois groups (1993), Preprint.
- [Be]
- K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237. MR 97m:11159
- [Bu-Mc]
- G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), 863-911. MR 84f:20005
- [Co]
- H. Cohen, A course in computational algebraic number theory, GTM 138, Springer-Verlag, Berlin, Heidelberg, New York, 1993. MR 94i:11105
- [Co-Di]
- H. Cohen and F. Diaz y Diaz, A polynomial reduction algorithm, Sém. Th. des Nombres Bordeaux (série 2) 3 (1991), 351-360. MR 93a:11107
- [Co-Di-Ol]
- H. Cohen, F. Diaz y Diaz and M. Olivier, Computing ray class groups, conductors and discriminants, Math. Comp. 67 (1998), 773-795. MR 98g:11128
- [Co-Hu-Mc]
- J. Conway, A. Hulpke and J. McKay, On transitive permutation groups, LMS J. Comput. Math., 1 (1998), 1-8.
- [Di-Ol]
- F. Diaz y Diaz and M. Olivier, Imprimitive ninth-degree number fields with small discriminants, Math. Comp. 64 (1995), 305-321. MR 95e:11153
- [Ei]
- Y. Eichenlaub, Problèmes effectifs de théorie de Galois en degrés à , Thèse, Université Bordeaux I, 1996.
- [Ei-Ol]
- Y. Eichenlaub and M. Olivier, Computation of Galois groups for polynomials with degree up to eleven (1996), Preprint.
- [Ga]
- F. Gassmann, Bemerkungen zur Vorstehenden Arbeit von Hurwitz, Math. Z. 25 (1926), 665-675.
- [Ha]
- H. Hasse, Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Physica-Verlag, Wien-Würtzburg, 1970. MR 42:1795
- [Je]
- A. Jehanne, Sur les extensions de
 à groupe de Galois  et  , Acta Arith. 69 (1995), 259-276. MR 95m:11126
- [Kw]
- S.-H. Kwon, Sur les discriminants minimaux des corps quaternioniens, Arch. Math. 67 (1996), 119-125. MR 97d:11165
- [Pe]
- R. Perlis, On the equation
 , J. Number Th. 9 (1977), 342-360. MR 56:5503
- [Po-Za]
- M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Math. and its Applications, Cambridge University Press, Cambridge, 1989. MR 92b:11074
- [Ro]
- X.-F. Roblot, Unités de Stark et corps de classes de Hilbert, C. R. Acad. Sci. Paris 323 (1996), 1165-1168. MR 97k:11162
- [Sm-Pe]
- B. de Smit and R. Perlis, Zeta functions do not determine class numbers, Bull. Am. Math. Soc. 31 (1994), 213-215. MR 95a:11100
- [Sm]
- G. W. Smith, Some polynomials over and their Galois groups (1993), Preprint.
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
11R37,
11Y40
Retrieve articles in all journals
with MSC (1991):
11R37,
11Y40
Additional Information
H. Cohen
Affiliation:
Laboratoire A2X, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
Email:
cohen@math.u-bordeaux.fr
F. Diaz y Diaz
Affiliation:
Laboratoire A2X, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
Email:
diaz@math.u-bordeaux.fr
M. Olivier
Affiliation:
Laboratoire A2X, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
Email:
olivier@math.u-bordeaux.fr
DOI:
http://dx.doi.org/10.1090/S0025-5718-99-01074-1
PII:
S 0025-5718(99)01074-1
Keywords:
Class field theory,
discriminant,
number field
Received by editor(s):
November 20, 1997
Posted:
February 24, 1999
Article copyright:
© Copyright 1999 American Mathematical Society
|
