Abstract
Let\(K = \mathbb{Q}(\sqrt {dm} ,\sqrt {dn} )\) be a biquadratic number field (where d,m,n∈ℤ, are uniquely determined); we say that it is monogenic if its ring of integers OK is of the form ℤ[θ]. We show that K is monogenic if and only if the two following conditions are satisfied:
-
(i)
2δm=2δn+4(2−δd) where δ=0 or 1 is defined by mn≡(−1)δ mod4;
-
(ii)
the equation (u2-v2)2(2δm)-(u2+v2)2(2δn)=±1 has solutions in ℤ.
We characterize all the imaginary monogenic biquadratic fieds and establishe other necessary conditions for monogenicity of real fields. Conjectures, numerical tables and statistics are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliographie
Faisant, A.: L'équation diophantienne du second degré, Hermann, no 1430, 1991
Gaal, I., Pethö, A. and Pohst, M.: On the resolution of index form equation in biquadratic Number Field, I, J. Number Theory38, 18–34 (1991)
Gaal, I., Pethö, A. and Pohst, M.: On the resolution of index form equation in biquadratic Number Field, II, J. Number Theory38, 35–51 (1991)
Gras, M.-N.: ℤ-bases d'entiers1, θ, θ2, θ3 dans les extensions cycliques de degré 4 de ℚ, Publ. Math. Fac. Sci. Besançon, Théorie des nombres, 11 pp. (1980/1981)
Gras, M.-N.: Non monogénéité des anneaux d'entiers, Séminaire de Théorie des Nombres (Univ. Bordeaux 1, Talence) exp. no15, 8 pp. (1985/86)
Györy, K.: Sur les générateurs des ordres monogènes des corps de nombres algébriques, Séminaire de Théorie des Nombres, (Univ. Bordeaux 1, Talence) exp. no32, 12 pp. (1983/84)
Henneman, F.: De vergelijking van Pell, Nieuw Tijdschr. Wisk.60, 1–30 (1972/73)
Kaplan, P. and Williams, K.: Pell's equationX 2-mY2=1,-4 and continued fractions, J. Number theory23, 169–182 (1986)
Kubota, K.: Über den bizyklischen biquadratischen zahlkörper, Nagoya Math. J.10, 65–85 (1956)
Nagell, Tr.: Contribution to the theory of a category of diophantine equations of the second degree with two unknowns, Nova Acta Reg. Soc. Scient. Upsaliensis, Ser. IV,16, 105–114 (1955)
Nagell, Tr.: On a special class of diophantine equations of the second degree, Ark. Math.3 no2, 1–12 (1953)
Nakahara, T.: On a power basis of the integer ring in an abelian biquadratic field, (in japanese), Surikaisekikenkyusho Kokyoroku371 (1983), 31–46. Experimental number theory (Proc. Sympo. Res. Inst. Math. Sci. Kyoto Univ, Kyoto (1971))
Nakahara, T.: On the indices and integral bases of non-cyclic but abelian biquadratic fields, Arch. Math.41 no6, 504–507 (1983)
Nakahara, T.: On cyclic biquadratic fields related to a problem of Hasse, Monasth. Math.94, 125–132 (1982)
Nakahara, T.: On the minimum index of a cyclic quartic field, Arch. Math.48, 322–325 (1987)
Stolt, B.: On a diophantine equation of the second degree, Ark. Mat.3 no33, 381–390 (1956)
Tanoé, F.: Monogénéité des corps biquadratiques, Thèse de Doctorat Université de Franche-Comté Besançon, Mention Mathématiques et applications, No d'ordre 141, 123 pp. (1990)
Williams, K.S.: Integer of biquadratic fields, Canad. Math. Bull.13 no4, 519–526. (1970)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gras, MN., Tanoé, F. Corps biquadratiques monogènes. Manuscripta Math 86, 63–79 (1995). https://doi.org/10.1007/BF02567978
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02567978