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Irreducible factors of a polynomial

Let n \geq 1 and let

    \[p_n(x)=x^{2^n}+x^{2^{n-1}}+1 \in \mathbb{Z}[x]\]

Find all irreducible factors of p_n(x).

Solution

Setting q_n(x)=x^{2^n}-x^{2^{n-1}}+1 we note that

    \[p_n(x)=p_{n-1}(x)q_{n-1}(x) \quad , \quad n \geq 2\]

Hence

(1)   \begin{equation*} p_n(x)=p_1(x)q_1(x)q_2(x) \cdots q_{n-1}(x)  \end{equation*}


It’s clear that p_1(x)=x^2+x+1 is irreducible over \mathbb{Z}. Now, for n \geq 1 let \Phi_n(x) be the n-th cyclotomic polynomial. Using well-known properties of \Phi_n, we have

    \[\Phi_{3 \cdot 2^n}(x)=\frac{\Phi_{2^n}(x^3)}{\Phi_{2^n}(x)}=\frac{x^{3 \cdot 2^{n-1}}+1}{x^{2^{n-1}}+1}=x^{2^n}-x^{2^{n-1}}+1=q_n(x)\]

Thus q_n is irreducible over \mathbb{Z} because cyclotomic polynomials are irreducible over \mathbb{Z}. Hence, by ( 1 ) p_n has exactly n irreducible factors and they are p_1, q_1, q_2, \dots , q_{n-1}.

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