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The monic quadratic polynomials in $\mathbb Z_2[x]$ are -

$x^2, x^2 + 1, x^2 + x, x^2 + x + 1$

  • $x^2 = x \cdot x$ so is reducible

  • $x^2 + 1 = (x + 1) \cdot (x + 1)$ so is reducible

  • $x^2 + x = x \cdot (x + 1)$ so is reducible

  • $x^2 + x + 1$ - The fact that this polynomial has no root in $\mathbb Z_2[x] $, that $\mathbb Z_2$ is a field, and that this polynomial has degree $\le 3$ implies that this polynomial is irreducible.

Have I got all that correct? In particular is the last claim correct or have I made an incorrect implication?

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    $\begingroup$ Apart from the fact that you claim that $\mathbb{Z}_2[x]$ is a field (rather than $\mathbb{Z}_2$ being one), the arguments are correct. $\endgroup$ – Tobias Kildetoft Nov 14 '13 at 12:04
  • $\begingroup$ Oops, I'll fix that, thanks. $\endgroup$ – sonicboom Nov 14 '13 at 13:32
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This is correct. Given the limited supply of polynomials of degree$~1$ (all monic), you could also say that you have already scrapped all products of two of them from your list, so what remains cannot be written as such a product.

Although this sieve method can in principle be used for any finite field and any degree polynomials, it becomes very laborious beyond $\Bbb F_2$ and say degree$~3$ polynomials. There are better methods for testing irreducibility over finite fields.

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