Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 4 / c / Solution

Put $Y=X^2$. Modulo two, $Y^2+9Y-2$ has the two simple roots zero and one, so Hensel lemma lifts them to roots $y_0,y_1\in {\mathbb{Z}}_2$, with $v_2(y_0)=1$ and $y_1\equiv 1(\mathrm{mod}8)$. The 2-adic unit criterion makes $y_1$ a square, so $X^2-y_1$ splits into two linear factors. The polynomial $X^2-y_0$ is Eisenstein and remains irreducible. Thus $X^4+9X^2-2$ has three irreducible factors, of degrees $1,1,2$, as in factorization of X4 plus 9X2 minus 2 over the 2-adic numbers.