Solution
= Solution
The defining roots are $\alpha,\alpha^2,\alpha^3,\alpha^4$. Frobenius conjugacy gives the two binary cyclotomic classes
$$
\{1,2,4,8\},
\qquad
\{3,6,12,9\}.
$$
Hence
$$
m_\alpha=m_{\alpha^2}=m_{\alpha^4}=X^4+X+1,
$$
while $\alpha^3$ has order five and
$$
m_{\alpha^3}=X^4+X^3+X^2+X+1.
$$
The generator is their least common multiple:
$$
\boxed{g(X)=X^8+X^7+X^6+X^4+1}.
$$