Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 74 / 4 / i / Solution

Use $R={\mathcal{O}}_{k_{\mathfrak{p}}}$, the valuation ring of the completion of a number field at a prime ideal, with uniformizer $\pi$ and residue field $\kappa$. This completeness is essential for coprime-factor Hensel lifting. Normalize the two residue factors to be monic first; if the original factors have reciprocal leading units, undo the normalization by multiplying the eventual factors by reciprocal lifts of those units.

Let the two monic residue degrees be $d_1,d_2$. Start with monic lifts $g_1,h_1$ satisfying $f\equiv g_1{h}_1(\mathrm{mod}\pi )$. Suppose monic lifts $g_r,h_r$ of these degrees satisfy $f-g_r{h}_r\in {\pi }^{r}R[X]$. Put $E_r=(f-g_r{h}_r)/{\pi }^r$, of degree less than $d_1+d_2$. Over $\kappa$, the linear mapis injective: a zero image makes ${\varphi }_1$ divide $u$, by coprimality, forcing $u=0$ by its degree bound, and then $v=0$. The dimensions agree, so this map is a bijection. Choose lifts $u_r,v_r$ whose image is ${\overline{E}}_r$, and setTheir product agrees with $f$ modulo ${\pi }^{r+1}$; the new cross term has factor ${\pi }^{2r}$ and $2r\ge r+1$. The degree bounds preserve monicity. Each coefficient is a Cauchy sequence in the complete discrete valuation ring $R$, so the limits $f_1,f_2$ satisfyIf ${\mathfrak{o}}_{\mathfrak{p}}$ were interpreted only as an ordinary localization rather than the completed valuation ring, the assertion would be false. For example $X^2-2$ splits modulo seven but is irreducible over $\mathbb{Q}$. The completion notation in the next part provides the intended local-field interpretation.