Proof that if poly $f \in \mathbb F_3$ of deg 4 or 5 with no roots is reducible then there is a monic irreducible polynomial of deg 2 dividing $f$ Since $\mathbb F_3[X]$ is a unique factorization domain there exists a unique factorization of irreducible elements. I know these must have degree 2 and 2 for $f$ of degree 4, and degree 2 and 3 for $f$ of degree 5. But how do i know any of these, or specifically the one with degree 2, are monic?

Leading coefficients in polynomials are like signs in prime numbers.
Positive prime numbers and monic irreducible polynomials are canonical representatives of irreducible elements.
Compare:


*

*Every integer has a unique factorization into positive prime numbers plus a sign:
$$
n = \pm \, p_1 \cdots p_m
$$

*Every polynomial over a field has a unique factorization into irreducible monic polynomials plus a leading coefficient:
$$
f = a \, p_1 \cdots p_m
$$
Uniqueness in both cases is up to order.
