So let's try applying your reasoning to the example given in the linked article. There it shows that in the integers extended by sqrt(-5) we have 2x3=(1-sqrt(5))x(1+sqrt(5)). So using your specific reasoning:
Let's take n=6
f1 : 2 x 3
f2 : (1-sqrt(5)) x (1+sqrt(5))
p : 2
m : 3
f2 does not contain p, ...
... and we cannot construct a prime p
from other primes or composites.
This means that f2 cannot be evenly
divisible by p, as that would require
it to have a prime factor of p which
it does not.
This is actually assuming (something equivalent to) the FTA. The example shows a case where f2 is evenly divisible by p, so your deduction here is wrong.
It is subtle.