The maximal unramified extensions of certain complex Abelian number fields
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2015), pp. 93-104.

Nous combinons les minorations des discriminants avec des considérations portant sur la ramification pour montrer, inconditionnellement, que le corps Q(-7,61) n’a pas d’extension non-ramifiée non-triviale (ce résultat a été montré par Yamamura avec l’aide de GRH). Cela rend inconditionnelle la détermination des extensions non-ramifiées maximales des coprs quadratiques complexes de nombre de classes 2. Sous GRH, nous montrons un résultat analogue pour le sous-corps de degré 14 de Q(ζ 49 ) (corps non étudié même sous GRH).

We combine root discriminant bounds with a ramification argument to show unconditionally that Q(-7,61) has no nontrivial unramified extension, a result first proved by Yamamura under the generalized Riemann hypothesis (GRH). This renders unconditional his determination of the maximal unramified extensions of the complex quadratic fields with class number 2. Assuming the GRH, we prove an analogous result for the degree 14 subfield of the cyclotomic field Q(ζ 49 ), a case previously not handled by conditional root discriminant bounds alone.

Reçu le :
Publié le :
DOI : 10.5802/pmb.14
Classification : 11R20, 11R21, 11R29, 20E22
Mots clés : Abelian fields, group extensions, root discriminants, unramified extensions
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Siman Wong. The maximal unramified extensions of certain complex Abelian number fields. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2015), pp. 93-104. doi : 10.5802/pmb.14. https://pmb.centre-mersenne.org/articles/10.5802/pmb.14/

[1] E. Artin, Galois theory, 2nd ed. University of Notre Dame Press, 1944. | Zbl

[2] J. E. Burns, The abstract definitions of groups of degree 8, Amer. J. Math. 37 (1915) 195-214. | DOI | MR | Zbl

[3] G. Butler and J. McKay, The transitive subgroups of degree up to eleven, Comm. Algebra 11 (1983) 863-911. | DOI | Zbl

[4] F. Diaz y Diaz, Tables minorant la racine m-ème du discriminant d’un corps de degré n, Publ. Math. d’Orsay, 1980. | Zbl

[5] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.4; 2014. http://www.gap-system.org

[6] F. Hajir and S. Wong, Specializations of one-parameter families of polynomials, Annales de l’Institut Fourier 56 (2006) 1127-1163. | DOI | MR | Zbl

[7] J. W. Jones and D. P. Roberts, A database of number fields. Preprint, May 2014. Accompanying searchable online database: http://hobbes.la.asu.edu/NFDB/

[8] S. Lang, Algebraic number theory, Springer-Verlag, 1986. | Zbl

[9] H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta. Arith. 24 (1974) 529-542. | DOI | MR | Zbl

[10] A. M. Odlyzko, On conductors and discriminants, in Algebraic number fields, A. Fröhlich, ed. 1977. | Zbl

[11] A. M. Odlyzko, Unpublished tables titled Discriminant bounds, November 29, 1976. Available from http://www.dtc.umn.edu/odlyzko/unpublished/index.html

[12] A. M. Odlyzko, Bounds for discriminants and related estimates for class numbers, regulators and zeros of zeta functions: a survey of recent results, Sém. Théorie des Nombres, Bordeaux 2 (1990) 119-141. | DOI | Numdam | MR | Zbl

[13] D. Roberts, Email communication. May 8, 2014.

[14] D. Robinson, A course in the theory of groups. 2nd ed. Springer-Verlag, 1996. | DOI

[15] J. P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972) 259-331. | DOI | Zbl

[16] K. Yamamura, On unramified Galois extensions of real quadratic number fields, Osaka J. Math. 23 (1986) 471-478. | Zbl

[17] K. Yamamura, The determination of the imaginary Abelian number fields with class number one. Math. Comp. 62 (1994) 899-921. | DOI | MR | Zbl

[18] K. Yamamura, The maximal unramified extension of the imaginary quadratic fields with class number two. J. Number Theory 60 (1996), no. 1, 42-50. | DOI | MR | Zbl

[19] K. Yamamura, On quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus field, Galois groups and modular forms (Saga, 2000). Kluwer Acad. Publ., 2003, p. 271-286. | DOI | Zbl

[20] K. Yamamura, Maximal unramified extensions of imaginary quadratic number fields of small conductors, II, J. Théor. Nombres Bordeaux 13 (2001) 633-649. | DOI | MR | Zbl

[21] K. Yamamura, Table of the imaginary Abelian number fields with class number one, http://tnt.math.se.tmu.ac.jp/pub/CDROM/CM-fields/imab.dvi | DOI | MR | Zbl

[22] K. Yamamura, Unpublished table titled Examples of real quadratic number fields with class number one having an unramified nonsolvable Galois extension.

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