Let $\mathbb{F}_q$ be a finite field. We have a polynomial ring $\mathbb{F}_q[t]$ and its field of fractions, which we denote $\mathbb{K}$. Suppose I have polynomials $f_1, \ldots, f_n$ in $\mathbb{K}[u]$. Let $g$ be the monic polynomial that is the gcd of $f_1, \ldots, f_n$.
Does it then follow that there exist $a_j \in \mathbb{K}[u] \ (1 \leq j \leq n)$ such that $$ a_1(u) f_1(u) + \cdots + a_n(u) f(n) = d(u)? $$
Does this follow for any polynomial ring over a field (in place of $\mathbb{K}$)? Thanks!
Yes. And you can find it inductively: first you find the gcd of the first two, then you use that gcd with the third polynomial to find the gcd for the first three and so on. You can then substitute backwards to find the $a_i$.