The one-parameter family of polynomials is a subfamily of the two-parameter family of Jacobi polynomials. We prove that for each , the polynomial is irreducible over for all but finitely many . If is odd, then with the exception of a finite set of , the Galois group of is ; if is even, then the exceptional set is thin.
La famille des polynômes à un seul paramètre est une sous-famille de la famille (à deux paramètres) des polynômes de Jacobi. On montre que pour chaque , quand on spécialise en , le polynôme est irréductible sur , sauf pour un nombre fini des valeurs . Si est impair, sauf pour un nombre fini des valeurs , le groupe de Galois de est ; si est pair, l’ensemble exceptionnel est mince.
@article{JTNB_2009__21_1_97_0, author = {Cullinan, John and Hajir, Farshid and Sell, Elizabeth}, title = {Algebraic properties of a family of {Jacobi} polynomials}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {97--108}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {1}, year = {2009}, doi = {10.5802/jtnb.659}, mrnumber = {2537705}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.659/} }
TY - JOUR AU - Cullinan, John AU - Hajir, Farshid AU - Sell, Elizabeth TI - Algebraic properties of a family of Jacobi polynomials JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 97 EP - 108 VL - 21 IS - 1 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.659/ DO - 10.5802/jtnb.659 LA - en ID - JTNB_2009__21_1_97_0 ER -
%0 Journal Article %A Cullinan, John %A Hajir, Farshid %A Sell, Elizabeth %T Algebraic properties of a family of Jacobi polynomials %J Journal de théorie des nombres de Bordeaux %D 2009 %P 97-108 %V 21 %N 1 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.659/ %R 10.5802/jtnb.659 %G en %F JTNB_2009__21_1_97_0
Cullinan, John; Hajir, Farshid; Sell, Elizabeth. Algebraic properties of a family of Jacobi polynomials. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 97-108. doi : 10.5802/jtnb.659. http://archive.numdam.org/articles/10.5802/jtnb.659/
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