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Let $f(x)$ be a polynomial in $(\mathbb{Z}/\mathbb{2Z})[x]$ of degree $2$ or $3$. Prove that $f(x)$ is irreducible if and only if $f(x)$ does not have a root in $\mathbb{Z}/\mathbb{2Z}.$

I know that $f(x)$ is irreducible if and only if $F[x]/(f(x))$ is a field.

Any suggestions/hints will be appreciated.

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    @ArturoMagidin Will do from now onwards. I didn't know that tag existed! I learned Abstract algebra from three different professors (and books) since undergrad days, so I am confused about notations and con$v$entions!2012-07-08

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Having a root in the field of coefficients is equivalent to having a linear factor. If a polynomial of degree 2 or 3 factors in a non-trivial way, then at least one of the factors is linear.

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    @JyrkiLahtonen Thanks! I forgot this falls directly from$a$know result.2012-07-08