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I want to show that $P(x)= x^4+2x^2+5$ is irreducible over rational numbers. I have decomposed the polynomial into $(x^2+ax+b)(x^2+cx+d)$, and since $P(x)$ is an even function, we have either $P(x)=(x^2+b)(x^2+d)$ or $P(x)=(x^2+ax+b)(x^2-ax+b)$. In the first case, $b$ and $d$ should be complex numbers. But in the second case, we have $2b-a^2=2$ and $b^2=5$, from which we can evaluate $a$ and $b$ to see that they are not rational numbers.

I'm new to the subject of irreducible polynomials. So I'm wondering if there are other methods to check if $P(x)$ is irreducible over rational numbers.

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    Does this answer your question? Prove that $x^4-2x^2-11$ is irreducible over $\Bbb Q$ You could use the Gauss Lemma. – Dietrich Burde May 27 '24 at 10:53
  • This polynomial is irreducible over $\mathbb Z_3[x]$ , hence irreducible over $\mathbb Q[x]$. – Peter May 27 '24 at 10:54
  • My first instinct when reading this was to let $x=\frac{p}{q}$ and then consider the three cases: $p,q$ both odd, $p$ even and $q$ odd or $p$ odd and $q$ even and then multiply through by $q^4$. You can then (maybe - this is the bit I haven't fully checked yet) whether all three lead to a case of $\text{even}=\text{odd}$ and therefore the polynomial has no rational roots by contradiction. – Red Five May 27 '24 at 11:09
  • @RedFive I think you mean the rational root theorem. But this does not allow to check whether we have two factors of degree $2$. – Peter May 27 '24 at 13:21
  • If $x^4 + 2 x^2 + 5 = ( x^2 + ax +b)(x^2 - ax + c)$ perhaps $a=0,$ but then $(b-c)^2 = -16.$ If $a \neq 0,$ then $b=c$ but then $b^2 = 5,$ and $b$ irrational. – Will Jagy May 27 '24 at 19:50
  • @Peter, correct. Which is why I didn't get very far. Two even powers makes for a more difficult problem. – Red Five May 27 '24 at 20:50
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    All its roots lie outside the unit circle (if $|z|\leq 1$ then $5=|z^4+2z^2|\leq |z|^4+2|z|^2\leq 3$), and constant coefficient is a prime, so it's irreducible. This has been here several times, e.g. in a form of Osada's criterion https://math.stackexchange.com/questions/3722135/is-x83x4-53-irreducible-over-mathbbzx/3722251#3722251. – Sil May 28 '24 at 13:49

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