Let \(f\in\FF_q[x]\) have degree at most \(4\). Find the smallest extension \(\FF_{q^N}\) that is guaranteed to split every such polynomial completely.
Solution
Every irreducible factor of a polynomial of degree at most \(4\) has degree in \[ \{1,2,3,4\}. \] An irreducible polynomial over \(\FF_q\) of degree \(d\) splits over \(\FF_{q^N}\) if and only if \(d\mid N\). Therefore every polynomial of degree at most \(4\) splits over \(\FF_{q^N}\) exactly when \[ 1,2,3,4\mid N. \] The smallest such integer is \[ N=\operatorname{lcm}(1,2,3,4)=12. \] Hence \[ \boxed{\FF_{q^{12}}} \] is the smallest extension that splits every polynomial of degree at most \(4\) over \(\FF_q\).
The degree \(12\) is minimal: irreducible polynomials of degrees \(3\) and \(4\) exist over every finite field, so any universal splitting field must have extension degree divisible by both \(3\) and \(4\), hence by \(12\).