Geodesic
On Fail-Rolle polynomials with few roots
Acta mathematica Universitatis Comenianae, Tome 79 (2010) no. 1 
L. H. Gallardo. On Fail-Rolle polynomials with few roots. Acta mathematica Universitatis Comenianae, Tome 79 (2010) no. 1. http://geodesic.mathdoc.fr/item/AMUC_2010_79_1_a8/
@article{AMUC_2010_79_1_a8,
     author = {L. H. Gallardo},
     title = {On {Fail-Rolle} polynomials with few roots},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2010},
     volume = {79},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2010_79_1_a8/}
}
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Voir la notice de l'article provenant de la source Comenius University

A splitting polynomial in one variable over a field is Fail-Rolle if its formal derivative does not split over the same field. It is known that the finite fields with more than four elements are exactly the finite fields for which there are Fail-Rolle polynomials. We describe all Fail-Rolle polynomials with at most five roots over a finite field of even characteristic.