Dans cette Note, nous considérons l'analogue dans les corps de fonctions du problème de Lehmer sur la fonction d'Euler. Soit et la fonction d'Euler de sur , où désigne un corps fini à q éléments. Nous montrons que si et seulement si (i) est irréductible, ou (ii) et est le produit de deux polynômes irréductibles non associés de degré 1, ou (iii) et est le produit de tous les polynômes irréductibles de degré 1, ou le produit de tous les polynômes irréductibles de degrés 1 et 2, ou le produit de trois polynômes irréductibles de degrés 1, 2 et 3, respectivement.
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let and be the Euler's totient function of over , where is a finite field with q elements. We prove that if and only if (i) is irreducible; or (ii) , is the product of any 2 non-associate irreducibles of degree 1; or (iii) , is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.
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@article{CRMATH_2017__355_4_370_0, author = {Ji, Qingzhong and Qin, Hourong}, title = {Lehmer's totient problem over $ {\mathbb{F}}_{q}[x]$}, journal = {Comptes Rendus. Math\'ematique}, pages = {370--377}, publisher = {Elsevier}, volume = {355}, number = {4}, year = {2017}, doi = {10.1016/j.crma.2017.03.007}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2017.03.007/} }
TY - JOUR AU - Ji, Qingzhong AU - Qin, Hourong TI - Lehmer's totient problem over $ {\mathbb{F}}_{q}[x]$ JO - Comptes Rendus. Mathématique PY - 2017 SP - 370 EP - 377 VL - 355 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.03.007/ DO - 10.1016/j.crma.2017.03.007 LA - en ID - CRMATH_2017__355_4_370_0 ER -
%0 Journal Article %A Ji, Qingzhong %A Qin, Hourong %T Lehmer's totient problem over $ {\mathbb{F}}_{q}[x]$ %J Comptes Rendus. Mathématique %D 2017 %P 370-377 %V 355 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.03.007/ %R 10.1016/j.crma.2017.03.007 %G en %F CRMATH_2017__355_4_370_0
Ji, Qingzhong; Qin, Hourong. Lehmer's totient problem over $ {\mathbb{F}}_{q}[x]$. Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 370-377. doi : 10.1016/j.crma.2017.03.007. http://www.numdam.org/articles/10.1016/j.crma.2017.03.007/
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