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Neighbour polynomials over finite field

Source: Romanian District Olympiad 2008, Grade XII, Problem 4

October 7, 2018
algebrapolynomialsuperior algebraRing Theoryfinite fieldWedderburn

Problem Statement

Let be a finite field K. K. Say that two polynoms f,g f,g from K[X] K[X] are neighbours, if the have the same degree and they differ by exactly one coefficient.
a) Show that all the neighbours of 1+X2 1+X^2 from Z3[X] \mathbb{Z}_3[X] are reducible in Z3[X]. \mathbb{Z}_3[X] .
b) If ∣K∣≥4, |K|\ge 4, show that any polynomial of degree ∣K∣−1 |K|-1 from K[X] K[X] has a neighbour from K[X] K[X] that is reducible in K[X], K[X] , and also has a neighbour that doesn´t have any root in K. K.
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