On the compositum of all degree d extensions of a number field
Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 655-672.

We study the compositum k [d] of all degree d extensions of a number field k in a fixed algebraic closure. We show k [d] contains all subextensions of degree less than d if and only if d4. We prove that for d>2 there is no bound c=c(d) on the degree of elements required to generate finite subextensions of k [d] /k. Restricting to Galois subextensions, we prove such a bound does not exist under certain conditions on divisors of d, but that one can take c=d when d is prime. This question was inspired by work of Bombieri and Zannier on heights in similar extensions, and previously considered by Checcoli.

Nous étudions le compositum k [d] de toutes les extensions de degré d d’un corps de nombres k dans une clôture algébrique fixée. Nous démontrons que k [d] contient toutes les sous-extensions de degré inférieur à d si et seulement si d4. Nous montrons que quand d>2, il n’existe pas de majorant c=c(d) sur le degré des éléments nécessaires pour engendrer les sous-extensions finies de k [d] /k. En se restreignant aux sous-extensions galoisiennes, nous montrons qu’un tel majorant n’existe pas sous certaines conditions sur les diviseurs de d, mais que l’on peut prendre c=d quand d est premier. Cette question a été inspirée par les travaux de Bombieri et Zannier sur les hauteurs dans des extensions similaires, et examinés par Checcoli.

DOI: 10.5802/jtnb.884
Classification: 12F10, 11R21, 20B05
Keywords: Number fields, infinite algebraic extensions, Galois theory, permutation groups
Itamar Gal 1; Robert Grizzard 2

1 Department of Mathematics, University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.
2 Department of Mathematics University of Texas at Austin 2515 Speedway, Stop C1200 Austin, TX 78712, U.S.A.
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     title = {On the compositum of all degree $d$  extensions of a number field},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     volume = {26},
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Itamar Gal; Robert Grizzard. On the compositum of all degree $d$  extensions of a number field. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 3, pp. 655-672. doi : 10.5802/jtnb.884. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.884/

[1] Y. Berkovich, Groups of prime power order, Vol. 1, Gruyter Expositions in Mathematics, 46, Walter de Gruyter GmbH & Co. KG, Berlin, (2008), with a foreword by Zvonimir Janko. | MR | Zbl

[2] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, 4 Cambridge University Press, Cambridge, (2006). | MR | Zbl

[3] E. Bombieri and U. Zannier, A note on heights in certain infinite extensions of , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, Mat. Appl., 9, 12, (2001), 5–14. | MR | Zbl

[4] G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11, 8, (1983), 863–911. | MR | Zbl

[5] S. Checcoli Fields of algebraic numbers with bounded local degrees and their properties, Trans. Amer. Math. Soc., 365, 4, (2013), 2223–2240. | MR | Zbl

[6] S. Checcoli and M. Widmer, On the Northcott property and other properties related to polynomial mappings, Math. Proc. Cambridge Philos. Soc., 155, 1, (2013), 1–12. | MR | Zbl

[7] S. Checcoli and U. Zannier, On fields of algebraic numbers with bounded local degrees, C. R. Math. Acad. Sci. Paris, 349, 1-2, (2011), 11–14. | MR | Zbl

[8] J. D. Dixon and B. Mortimer, Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, New York, (1996). | MR | Zbl

[9] K. Doerk and T. Hawkes, Finite soluble groups, volume 4 of de Gruyter Expositions in Mathematics, Walter de Gruyter & Co., Berlin, (1992). | MR | Zbl

[10] D. S. Dummit and R. M. Foote, Abstract algebra, John Wiley & Sons Inc., Hoboken, NJ, third edition, (2004). | MR | Zbl

[11] W. Feit, Some consequences of the classification of finite simple groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., 37, (1980), Amer. Math. Soc., Providence, R.I., 175–181. | MR | Zbl

[12] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, The Clarendon Press Oxford University Press, New York, fifth edition, (1979). | MR | Zbl

[13] A. Hulpke, Transitive permutation groups - A GAP data library www.gap-system.org/Datalib/trans.html.

[14] D. W. Masser, The discriminants of special equations, Math. Gaz., 50, 372, (1966), 158–160.

[15] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, (2008). | MR | Zbl

[16] O. Neumann, On the imbedding of quadratic extensions into Galois extensions with symmetric group, in Proceedings of the conference on algebraic geometry (Berlin, 1985), Teubner-Texte Math., 92, (1986), Leipzig, Teubner, 285–295. | MR | Zbl

[17] V. V. Prasolov, Polynomials, volume 11 of Algorithms and Computation in Mathematics, Springer-Verlag, Berlin, (2004), translated from the 2001 Russian second edition by Dimitry Leites. | MR | Zbl

[18] J.-P. Serre, Topics in Galois theory, volume 1 of Research Notes in Mathematics, Jones and Bartlett Publishers, Boston, MA, (1992). Lecture notes prepared by Henri Damon [Henri Darmon], With a foreword by Darmon and the author. | MR | Zbl

[19] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.5.4, (2012).

[20] M. Widmer, On certain infinite extensions of the rationals with Northcott property, Monatsh. Math., 162, 3, (2011), 341–353. | MR | Zbl

[21] H. Wielandt, Finite permutation groups, translated from the German by R. Bercov. Academic Press, New York, (1964). | MR | Zbl

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