In my course in p-adic numbers I need to prove that any factorization mod $p$ in relatively prime factors lifst to $\mathbb{Z}_p$. One part of the question is the following:
Given are two polynomials $A$ and $B$ in $\mathbb{F}_p[x]$ of degree $a$ and $b$ respectively, and $A$ and $B$ are coprime. Also we have a polynomial $R \in \mathbb{F}_p[x]$ of degree less or equal than $a+b-1$. Now I need to show that there exist $u,v \in \mathbb{F}[x]$ of degrees less than $b$ respectively $a$ such that $uA+vB=R$ in $\mathbb{F}_p[x]$.
I already noticed that without loss of generality we can reduce the problem by looking at $R=x^m$ for some $m \leq a+b-1$. Also so $A$ and $B$ are coprime, we can write $uA+vB=1$ for some $u$ and $v$, but when we multiply through by $R$, the degrees are way too high. How to proceed?