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

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.

Set $L=K_{\mathfrak{P}}$ and $F=k_{\mathfrak{p}}$, with valuation rings $S,R$. Let $e$ be the ramification index, $f$ the residue-field degree, and $p$ the residue characteristic. A base uniformizer $\pi$ generates ${\mathfrak{P}}^e$ in $S$, up to a unit. The free $R$-module $S$ has rank $ef$. For an integral $a\in S$, its field trace is the matrix trace of multiplication by $a$ on this module. Reduction modulo $\pi$ preserves the matrix trace.

The quotient $S/\pi S$ has a filtrationEach of the $e$ successive quotients is a one-dimensional vector space over ${\kappa }_L=S/\mathfrak{P}$. On each, multiplication by $a$ acts as multiplication by its residue $\bar{a}$, whose trace over ${\kappa }_F=R/\pi R$ is ${\mathrm{Tr}}_{{\kappa }_L/{\kappa }_F}(\bar{a})$. A basis adapted to the filtration makes the multiplication matrix block triangular, so its trace is the sum of these quotient traces. This proves trace reduction in a ramified local extension:Since the hypothesis says $p\mid e$, the right side is zero in ${\kappa }_F$. ThusNo normality hypothesis is needed. The original PDF only requires divisibility of $e$, not divisibility exactly once; the converted TeX has corrupted that condition and several local subscripts.