# Prove the polynomial is irreducible on the prime field $F_2$

How to prove that $$x^{2^n}+x+1$$ is irreducible in $F_2$

-Is this question relevant to finite field?

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@CalvinLin It might still be the product of two smaller degree polynomials (which also have no roots in $\mathbb F_2$). –  Hagen von Eitzen Dec 31 '12 at 11:35
You might be interested in this math.stackexchange.com/questions/122274/… –  Ram Dec 31 '12 at 11:37
@HagenvonEitzen Yes, I realized that soon after, when I tried factoring it. –  Calvin Lin Dec 31 '12 at 11:39
You can't prove it because it's not true: $x^8+x+1 = (x^2+x+1)(x^6+x^5+x^3+x^2+1)$.