Question about proving irreducibility in $\mathbb{F}_2$

Question

For the polynomial $x^3 + x + 1$ and trying to prove that it is irreducible over the field $\mathbb{F}_2$, I was told that since $f$ is a degree 3 polynomial that it is simply enough to verify that there are no roots in $\mathbb{F}_2$.

What does this have to do with irreducibility? The calculation is simple enough but I don't understand why this verification can show irreducibility.

A polynomial $f$ over a field $K$ of degree $3$ is irreducible if and only if it has no root. — Dietrich Burde, Apr 06 '21 at 18:17

score 0 · Answer 1 · answered Apr 06 '21 at 18:09

0

This is because if a cubic polynomial is reducible, then one of its factors must be linear. For polynomials of degree four or higher, it doesn't work.

answered Apr 06 '21 at 18:09

TonyK

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Question about proving irreducibility in $\mathbb{F}_2$

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