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I want to show that $f(x) = 7x^4 + 103x^3 + 802x^2 + 6x + 75$ is irreducible in $\mathbb{Q}[x]$. My attempts are:

Consider $f(x)$ reduced mod $2$. This gives $x^4 + x^3 + 1$ which is irreducible as it has no root and is not divisible by $x^2 + x + 1$. So we know that $f(x)$ has no integer roots. Now suppose that $f(x) = g_1(x) g_2(x)$ for $g_1,g_2 \in \mathbb{Z}[x]$. Considering the reduction mod $2$ yields a contradiction so we have that $f(x)$ is irreducible in $\mathbb{Z}[x]$.

I would like to apply now Gauss' Lemma, but the coefficients are not coprime, so I can't. Can someone give an alternative approach?

joan
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  • What you did is correct. Also you do not need to have coprime coefficients to apply Gauss's Lemma. Thus, you are actually done! However, I am curious why do you thought that coprimeness of the coefficients are needed? – ALNS Apr 28 '20 at 16:33
  • I do think that it is required that the coefficients are coprime. – joan Apr 28 '20 at 16:43
  • https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomial) – joan Apr 28 '20 at 16:43
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    The gcd is here equal to $1$. Recall, it is not pairwise coprime required. Since already $gcd(7,103)=1$, we are done. – Dietrich Burde Apr 28 '20 at 16:46
  • I understood the definition wrongly then! Using Gauss Lemma the result now follows. – joan Apr 28 '20 at 17:01

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