I would like to show that $P(X)=X^4-20X^2+16$ is irreducible on $\mathbb{Q}$, how to proceed ?
2026-04-13 08:25:09.1776068709
Show that a polynomial is irreducible on $\mathbb{Q}$
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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.