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WILD RAMIFICATION IN TRINOMIAL EXTENSIONS AND GALOIS GROUPS

Part of: General field theory Field extensions Algebraic number theory: local and $p$-adic fields Algebraic number theory: global fields

Published online by Cambridge University Press:  12 March 2020

BOUALEM BENSEBAA ,
ABBAS MOVAHHEDI  and
ALAIN SALINIER
BOUALEM BENSEBAA
Affiliation:
Laboratoire LA3C, Faculté de Mathématiques, USTHB, Bab Ezzouar16111, Algeria, e-mail: b.benseba@usthb.dz
ABBAS MOVAHHEDI
Affiliation:
XLIM (UMR 7252 CNRS, Université de Limoges), 87060LimogesCedex, France, e-mails: abbas.movahhedi@unilim.fr, alain.salinier@unilim.fr
ALAIN SALINIER
Affiliation:
XLIM (UMR 7252 CNRS, Université de Limoges), 87060LimogesCedex, France, e-mails: abbas.movahhedi@unilim.fr, alain.salinier@unilim.fr
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Abstract

It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

MSC classification

Primary: 11R32: Galois theory 12F10: Separable extensions, Galois theory
Secondary: 11S15: Ramification and extension theory 12E10: Special polynomials
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 106 - 120
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

REFERENCES

Abhyankar, S. S., Galois theory on the line in nonzero characteristic, Bull. Amer. Math. Soc. 27(1) (1992), 68–133.CrossRefGoogle Scholar
Bensebaa, B., Movahhedi, A. and Salinier, A., The Galois group of X p + aX s + a, Acta Arith. 134(1) (2008), 55–65.CrossRefGoogle Scholar
Bensebaa, B., Movahhedi, A. and Salinier, A., The Galois group of X p + aX p−1 + a , J. Number Theory 129(4) (2009), 824–830.CrossRefGoogle Scholar
Bourbaki, N., Eléments de Mathématique, Algèbre, Chapitres 4 à 7 (Masson, Paris, 1981).Google Scholar
Bouyacoub, C. and Salinier, A., On the solvability of an Eisenstein trinomial of prime degree, Acta Arith. 178(4) (2017), 385–396.CrossRefGoogle Scholar
Cohen, S. D., Movahhedi, A. and Salinier, A., Double transitivity of Galois groups of trinomials, Acta Arith. 82(1997), 1–15.CrossRefGoogle Scholar
Cohen, S. D., Movahhedi, A. and Salinier, A., Galois groups of trinomials, J. Alg. 222(1999), 561–573.CrossRefGoogle Scholar
Cohen, S. D., Movahhedi, A. and Salinier, A., Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47(2000), 173–196.CrossRefGoogle Scholar
Dixon, J. D. and Mortimer, B., Permutation Groups, Graduate Texts in Mathematics, vol. 163 (Springer-Verlag, New-York, 1996).CrossRefGoogle Scholar
Feit, W., Some consequences of the classification of finite simple groups, Proc. Sympos. Pure Math. 37 (1980), 175–181.CrossRefGoogle Scholar
Gauckler, L., The Galois group of the Eisenstein polynomial X 5 + aX + a , Arch. Math. 90 (2008), 136–139.Google Scholar
Jensen, C. U., Ledet, A. and Yui, N. Generic polynomials. Constructive aspects of the inverse Galois problem, Mathematical Sciences Research Institute Publications, vol. 45 (Cambridge University Press, Cambridge, 2002).Google Scholar
Kölle, M. and Schmid, P., Computing Galois groups by means of Newton polygons, Acta Arith. 115 (2004), 71–84.CrossRefGoogle Scholar
Komatsu, K., Discriminants of certain algebraic number fields, J. Reine Angew. Math. 285 (1976), 114–125.Google Scholar
Komatsu, K., On the Galois group of x p + p tb(x + 1) = 0, Tokyo J. Math. 15 (1992), 351–356.CrossRefGoogle Scholar
Lalesco, T., Sur le groupe des équations trinômes, Bulletin de la Société Mathématique de France 35 (1907), 75–76.CrossRefGoogle Scholar
Llorente, P., Nart, E. and Vila, N., Discriminants of number fields defined by trinomials, Acta Arith. XLIII (1984), 367–373.CrossRefGoogle Scholar
Llorente, P., Nart, E. and Vila, N., Decomposition of primes in number fields defined by trinomials, Sém. Théor. Nombres Bordeaux (2) 3(1) (1991), 27–41.CrossRefGoogle Scholar
Movahhedi, A., Galois group of X p + aX + a, J. Alg. 180(3) (1996), 966–975.CrossRefGoogle Scholar
Movahhedi, A. and Salinier, A., The primitivity of the Galois group of a trinomial, J. London Math. Soc. (2) 53 (1996), 433–440.CrossRefGoogle Scholar
Narkiewicz, W., Elementary and analytic theory of algebraic numbers, 2nd edition (Springer-Verlag, Berlin; PWN-Polish Scientific Publishers, Warsaw, 1990).Google Scholar
Osada, H., The Galois groups of the polynomials x n + ax s + b. II, Tohoku Math J. (2) 39 (1987), 437–445.Google Scholar
Plans, B. and Vila, N., Trinomial extensions of $\mathbb{Q}$ with ramification conditions, J. Number Theory 105 (2004), 387–400.CrossRefGoogle Scholar
Roland, G., Yui, N. and Zagier, D., A parametric family of quintic polynomials with Galois group D 5 , J. Number Theory 15(1982), 137–142.CrossRefGoogle Scholar
Spearman, B. K. and Williams, K. S., On solvable quintics X 5 + aX + b and X 5 + aX 2 + b, Rocky Mountain J. Math. 26(2) (1996), 753–772.CrossRefGoogle Scholar
Swan, R. G., Factorization of polynomials over finite fields, Pacific J. Math. 12(1962), 1099–1106.CrossRefGoogle Scholar
Uchida, K., Unramified extensions of quadratic number fields. II, Tôhoku Math. J. (2) 22 (1970), 220–224.CrossRefGoogle Scholar
Weber, H., Lehrbuch der Algebra (Braunschweig Druck und Verlag von Friedrich Vieweg und Sohn, 1898).Google Scholar