unique factorization without fundamental theorem?

Question

For an arbitrary polynomial $P(x)$ over ${\bf C}$, can it be shown that IF $P(x)$ can be factored, THEN the factorization is unique, without using the fundamental theorem of algebra, and only elementary methods (for example, not analytic continuation)?

Answer 1

Unique factorization does not need analysis or the fundamental theorem of algebra: when $K$ is any field at all (not just the complex numbers), $K[x]$ has unique factorization. This is proved in many abstract algebra textbooks.

A more technical result, also proved in many abstract algebra textbooks, is that if the integral domain $A$ has a unique factorization then $A[x]$ has unique factorization. But the proof when $A$ is a field is much simpler (because $K[x]$, as mentioned in a comment below, is in fact Euclidean when $K$ is a field, but it turns out that $A[x]$ is not Euclidean when $A$ is an integral domain that's not a field).