Show that a polynomial is irreducible on $\mathbb{Q}$

Question

I would like to show that $P(X)=X^4-20X^2+16$ is irreducible on $\mathbb{Q}$, how to proceed ?

See this post. The proof is similar. – Dietrich Burde May 07 '21 at 20:56 — Dietrich Burde, May 07 '21 at 20:56
$\Delta=10^2-16=84≠n^2, n\in\mathbb Z$ – lone student May 07 '21 at 21:05 — lone student, May 07 '21 at 21:05

score 1 · Answer 1 · answered May 07 '21 at 20:59

1

By the rational root theorem, $P$ has no rational root. So we only need to study a possible factorization with integer coefficients $$ P(x)=(x^2+ax+b)(x^2+cx+d) $$ Comparison with the coefficients gives equations over the integers with no solution.

answered May 07 '21 at 20:59

Dietrich Burde

130,978

This is a quadratic equation respect to $X^2$ isn't it? So, is it enough to show that $X^2 $ (after solving quadratic) is irrational? Because, If $X^2$ is irrational, then $X$ irrational. Am I right? Thank you. – lone student May 07 '21 at 21:02
You mean like egreg's answer in the duplicate? – Dietrich Burde May 07 '21 at 21:04

Show that a polynomial is irreducible on $\mathbb{Q}$

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