Non-primitive number fields of degree eight and of signature (2,3), (4,2), and (6,1) with small discriminant

Selmane, Schehrazad

doi:10.1090/S0025-5718-99-00998-9

Non-primitive number fields of degree eight and of signature $(2,3)$, $(4,2)$, and $(6,1)$ with small discriminant
HTML articles powered by AMS MathViewer

by Schehrazad Selmane PDF

Math. Comp. 68 (1999), 333-344 Request permission

Abstract:

We give the lists of all non-primitive number fields of degree eight having two, four and six real places of discriminant less than 6688609, 24363884 and 92810082, respectively, in absolute value. For each field in the lists, we give its discriminant, the discriminant of its subfields, a relative polynomial generating the field over one of its subfields and its discriminant, the corresponding polynomial over $\mathbb {Q}$, and the Galois group of its Galois closure.

References

H. Anai, M. Noro, and K. Yokoyama, Computation of the splitting fields and the Galois groups of polynomials, Algorithms in algebraic geometry and applications (Santander, 1994) Progr. Math., vol. 143, Birkhäuser, Basel, 1996, pp. 29–50. MR 1414444
Ch. Batut, D. Bernadi, H. Cohen and M. Olivier, GP/PARI Calculator version 1.37, Publ. Université Bordeaux 1 (1991).
Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. MR 695893, DOI 10.1080/00927878308822884
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, DOI 10.5802/jtnb.55
Francisco Diaz y Diaz, Tables minorant la racine $n$-ième du discriminant d’un corps de degré $n$, Publications Mathématiques d’Orsay 80 [Mathematical Publications of Orsay 80], vol. 6, Université de Paris-Sud, Département de Mathématiques, Orsay, 1980 (French). MR 607864
F. Diaz y Diaz, Private communication to the author.
F. Diaz y Diaz, Petits discriminants des corps de nombres totalement imaginaires de degré $8$, J. Number Theory 25 (1987), no. 1, 34–52 (French, with English summary). MR 871167, DOI 10.1016/0022-314X(87)90014-X
F. Diaz y Diaz and M. Oliver, Corps imprimitifs de degré $9$ de petit discriminant, Preprint.
H. J. Godwin, On quartic fields of signature one with small discriminant, Quart. J. Math. Oxford Ser. (2) 8 (1957), 214–222. MR 97375, DOI 10.1093/qmath/8.1.214
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Armin Leutbecher, Euclidean fields having a large Lenstra constant, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 2, 83–106 (English, with French summary). MR 786536, DOI 10.5802/aif.1011
J. Martinet, Méthodes géométriques dans la recherche des petits discriminants, Sém. de Théorie des nombres de Paris 1983/84, Birkhäuser Verlag, Basel (1985), 147–179.
Jacques Martinet, Petits discriminants des corps de nombres, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 151–193 (French). MR 697261
M. Noro and T. Takeshima, Risa/Asir—a computer algebra system, Proc. ISSAC 92, ACM Press, 1992, pp. 387–396.
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. MR 644904, DOI 10.1016/0022-314X(82)90061-0
M. Pohst, J. Martinet, and F. Diaz y Diaz, The minimum discriminant of totally real octic fields, J. Number Theory 36 (1990), no. 2, 145–159. MR 1072461, DOI 10.1016/0022-314X(90)90069-4
M. Pohst, On computing isomorphisms of equation orders, Math. Comp. 48 (1987), no. 177, 309–314. MR 866116, DOI 10.1090/S0025-5718-1987-0866116-2
Georges Poitou, Sur les petits discriminants, Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 1, Secrétariat Math., Paris, 1977, pp. Exp. No. 6, 18 (French). MR 551335
A. Valibouze, Théorie de Galois constructive, Mémoire d’Habilitation à Diriger les Recherches, Université Pierre et Marie Curie, 1994.