Why is there a $q_i$ such that $q_j|q_i$?

Question

Let $q_i$, a sequence of of irreducible polynomials where $q_i$'s highest-order term has coefficient $c_n = 1$ (by the way, what's the right term to describe this property?)

Anyhow, let's look at:

$$ \prod_{i\in I, i\ne j} q_i^{m_i} = q_j^{n_j-m_j} \prod_{i\in I, i\ne j} q_i^{n_i}$$

Now, since our ring is $R=F[X]$, $q_j$ is prime (I am not fully sure what does it mean for a polynomial to be prime). Therefore, there's a $q_i\ne q_j$ such that $q_j|qi$.

So in conclusion, I'd be glad to get clarifications for:

1. Terminology: What's the right definition for a polynomial where it's highest-order term has coefficient $c_n = 1$.
2. I am used to think of primes as numbers. I know the formal definition is $q$ is prime iff for every $a,b\in R$: $q|ab \implies q|a \lor q|b$. Could you give an example of such polynomial in $F[X]$?
3. Why is there a $q_i$ dividing $q_j$?

Thanks.