We know that one irreducible polynomial on $\Bbb{Z}_2[x]$ is $x^8 + x^4 + x^3 + x + 1$. How to check that it is irreducible? And how to generate irreducible polynomial for any degree?
2026-03-26 06:30:51.1774506651
Irreducible Polynomial of Galois field
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Hint: The polynomial $x^{p^n}-x$ is the product of all monic irreducible polynomials in ${\Bbb F}_p[x]$ whose degree is a divisor of $n$. In this way, one could iteratively construct irreducible polynomials of larger and larger degree.