Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

In this paper, we show that the truncated binomial polynomials defined by (P_{n,k}(x)={sum }_{j=0}^{k} {n choose j} x^{j}) are irreducible for each k≤6 and every n≥k+2. Under the same assumption n≥k+2, we also show that the polynomial P _n,k cannot be expressed as a composition P _n,k (x) = g(h(x)) with (g in mathbb {Q}[x]) of degree at least 2 and a quadratic polynomial (h in mathbb {Q}[x]). Finally, we show that for k≥2 and m,n≥k+1 the roots of the polynomial P _m,k cannot be obtained from the roots of P _n,k , where m≠n, by a linear map.