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f(x) = (x -a_1)(x - a_2) ... (x -a_n) - 1,not divisible

Source: Nordic Mathematical Contest 1992 #2

October 5, 2017
Integer Polynomialpolynomial divisionalgebra

Problem Statement

Let n>1n > 1 be an integer and let a1,a2,...,ana_1, a_2,... , a_n be nn different integers. Show that the polynomial f(x)=(x−a1)(x−a2)⋅...⋅(x−an)−1f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1 is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than nn and such that the highest power of xx has coefficient 11.
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