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Question: Find the irreducible decomposition of $$x^{16}+x$$ over $\mathbb{F_8}=\mathbb{F_2}[t]/(t^{3}+t+1)$.

It's easy to show that $$x^{16}+x=x(x+1)(x^{2}+x+1)(x^{4}+x+1)(x^{4}+x^{3}+1)(x^{4}+x^{3}+x^{2}+x+1)$$in $\mathbb{F_2}[x]$, but how to get the irreducible decomposition of $$(x^{4}+x^{3}+1)(x^{4}+x+1)(x^{4}+x^{3}+x^{2}+x+1)$$ over $\mathbb{F_8}$? Thanks a lot.

user26857
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llll
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1 Answers1

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Each over $\mathbb F_2$ irreducible polynomial of degree $4$ is irreducible over $\mathbb F_8$ as well. The reason is quite easy: A polynomial of degree $4$ would factor over $\mathbb F_{2^4}=\mathbb F_{16}$ into linear factors. Since $\mathbb F_{16} \cap\mathbb F_{8}=\mathbb F_{2}$ no factorisation of such a polynomial is visible in $\mathbb F_8$.

principal-ideal-domain
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