irreducible polynomial of degree less than 3 and root
Proposition
Let \(f \in R[X]\) be a primitive polynomial with \(\mathrm{deg}(f) \leq 3\) for a commutative ring \(R\).
TFAE:
- \(f\) is irreducible
- \(f\) has no root
Proof
1) \(\implies\) 2)
2) \(\implies\) 1)
Suppose \(f = g \cdot h\).
Then it follows w.l.o.g. that \(\mathrm{deg}(g) = 3\) as otherwise one of \(g,h\) would be a linear polynomial.
Then \(h \in R^{\times}\) since \(f\) is a primitive polynomial.