In the book p-adic Differential Equations by Kiran.S.Kedlaya one finds the exercise II.(3):
If $F$ is a complete nonarchimedean field, $P(X)\in\mathfrak o_F[X]$, and the reduction of $P(X)$ in $k_F$ factors as $\overline Q\cdot\overline R$, with $\overline Q$, $\overline R$ co-prime, then there exists a unique factorisation $P=QR$ in $\mathfrak o_F[X]$ with $Q,R$ lifting $\overline Q,\,\overline R$.
The hint says
Since $\overline Q$, $\overline R$ are co-prime, there exist $\overline S, \overline T\in k_F[X]$ with $\overline Q\overline S+\overline R\overline T=1$, $\deg(\overline S)<\deg(\overline R)$ and $\deg(\overline T)<\deg(\overline Q)$. Use lifts of these to fulfill the conditions of the master factorisation theorem (stated at the end of the question).
Attempt:
We need to choose three elements $a,b,c\in\mathfrak o_F[X]$ and three complete additive subgroups $U,V,W$ satisfying the conditions. Since we want to factorise $P$, we shall choose $c=P$. Then choose arbitrary lifts $Q,R,S,T$ and let $a=Q$ and $b=R$. Since we shall have $ab-c\in W$, and we only know that $ab-c\in\left<p\right>$, we then choose $W=\left<p\right>\subseteq\mathfrak o_F[X]$.
Next let $U=\left<pT\right>$ and $V=\left<pS\right>$. As $QS+RT\equiv1\pmod p$, it follows that $f(u,v):=av+bu$ is a surjection from $U\times V$ to $W$.
It remains to show that there exists $\lambda>0$ such that $$\left|f(u,v)\right|\geq\lambda\cdot\text{max}\left\{\left|a\right|\left|v\right|,\left|b\right|\left|u\right|\right\}$$ and $$\left|ab-c\right|<\lambda^2\left|c\right|.$$
But I have no clue how to find such a $\lambda$, other than obviously that $\lambda\leq1$. And I do not see a way of applying the degree restrictions on $\overline Q$ and $\overline R$.
I have thought about varying the lifts $Q,R$ by some multiples of $p$ so that we can choose $\lambda=1$, but to no avail. And I have run out of ideas now.
Master factorisation theorem:
Let $R$ be a nonarchimedean ring. Suppose there are three elements $a,b,c\in R$ and three additive subgroups $U,V,W\subseteq R$ such that
- The space $U,V$ are complete under the norm and $U\cdot V\subseteq W$.
- The map $f(u,v):=av+bu$ is a surjection from $U\times V$ to $W$.
- There exists $\lambda>0$ such that $$\left|f(u,v)\right|\geq\lambda\cdot\text{max}\left\{\left|a\right|\left|v\right|,\left|b\right|\left|u\right|\right\},\,\forall u\in U, v\in V.$$
- We have $ab-c\in W$ and $\left|ab-c\right|<\lambda^2\left|c\right|$.
Then there exists a unique pair $(x,y)\in U\times V$ such that $$c=(a+x)(b+y),\quad\left|x\right|<\lambda\left|a\right|,\ \left|x\right|<\lambda\left|a\right|.$$ We also have $$\left|x\right|\leq\lambda^{-1}\left|ab-c\right|\left|b\right|^{-1}\quad\left|y\right|\leq\lambda^{-1}\left|ab-c\right|\left|a\right|^{-1}.$$
Any help or reference is sincerely appreciated. Thanks in advance.