Can you have a degree 7 irreducible polynomial over \(\mathbb{F}_p\)? How about a degree 14 irreducible polynomial?
Solution
Yes in both degrees; in fact, over every finite field \(\mathbb F_p\) there are irreducible polynomials of every positive degree.
For \(n\ge1\), let \(\mathbb F_{p^n}\) be the field with \(p^n\) elements. Its multiplicative group is cyclic. If \(\alpha\) generates \(\mathbb F_{p^n}^{\times}\), then \[ \mathbb F_p(\alpha)=\mathbb F_{p^n}, \] because the left side is a subfield containing every nonzero power of \(\alpha\), hence every element of \(\mathbb F_{p^n}\). Therefore the minimal polynomial of \(\alpha\) over \(\mathbb F_p\) has degree \[ [\mathbb F_p(\alpha):\mathbb F_p]=n. \] So irreducible polynomials of degrees \(7\) and \(14\) exist.
The number \(N_p(n)\) of monic irreducible polynomials of degree \(n\) is \[ N_p(n)=\frac1n\sum_{d\mid n}\mu(d)p^{n/d}. \] Hence \[ N_p(7)=\frac{p^7-p}{7}>0 \] and \[ N_p(14)=\frac{p^{14}-p^7-p^2+p}{14}>0. \]