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How can one show that the polynomial $$x^4+x^3+3$$ is irreducible in $\Bbb{Q}[x]$? I only know Eisenstein criterion, but it doesn't apply directly. Of course I verified that there are no rational roots - but how can I exclude that it's a product of two irreducible quadratic polynomials?

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up vote 5 down vote accepted

A simple approach is to remark that $x^4+x^3+3$ is irreducible modulo $2$. Indeed, $x^4+x^3+1\pmod{2}$ has no roots in $\mathbb{F}_2$ and is not divisible by $x^2+x+1$.

Thus, your polynomial is also irreducible over the rationals.

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