In this paper, we show that the truncated binomial polynomials defined by $P_{n, k}(x)=sum^k_{j=0}({nchoose j})x^j$ are irreducible for each $kleq 6$ and every $ngeq k+2$. Under the same assumption $ngeq k+2$, we also show that the polynomial $P_{n, k}$ cannot be expressed as a composition $P_{n, k}(x)=g(h(x))$ with $ginmathbb{Q}[x]$ of degree at least 2 and a quadratic polynomial $hinmathbb{Q}[x]$. Finally, we show that for $kgeq 2$ and $m, ngeq k+1$ the roots of the polynomial $P_{m, k}$ cannot be obtained from the roots of $P_{n, k}$, where $m eq n$, by a linear map.
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机译：在本文中，我们证明了由$ P_ {n，k}（x）= sum ^ k_ {j = 0}（{nchoose j}）x ^ j $定义的截断的二项式多项式对于每个$ kleq 6 $都是不可约的。以及每个$ ngeq k + 2 $。在相同的假设$ ngeq k + 2 $下，我们还表明多项式$ P_ {n，k} $不能表示为组成$ P_ {n，k}（x）= g（h（x））$ $ ginmathbb {Q} [x] $的度数至少为2，并且二次多项式$ hinmathbb {Q} [x] $。最后，我们表明对于$ kgeq 2 $和$ m，ngeq k + 1 $，无法从$ P_ {n，k} $的根获得多项式$ P_ {m，k} $的根，其中$ m eq n $，通过线性映射。
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