Let $p$ be a prime number and let $\zeta_{p}$ be the pth roots of unity and le $\mathbb{Q}(\zeta_{p})/\mathbb{Q}$ be a Galois extension and $G=Gal(\mathbb{Q}(\zeta_{p})/\mathbb{Q})$ be its Galois group we know that $|G|=\phi(p)=p-1 $ And let $S=\{1,\zeta,\zeta^{2},.....,\zeta^{p-2}\}$ be a basis of the Galois extension $\mathbb{Q}(\zeta_{p})/\mathbb{Q}$ and let $m_{\alpha}(x)=\alpha x $ be a linear operator .and $[m_{\alpha}]$ be its matrix representation
Edit : $\alpha=1+a_{1}\zeta$
My question whether any formula for $det([m_{\alpha}])$ or at least an approximation of determinant matrix of cyclotomic extension ?
Hint 1: What is the minimal polynomial of $m_{\alpha}$? How does the minimal polynomial of a linear operator relate to its characteristic polynomial?
Hint 2: Let $p$ denote the polynomial $$ p(x) = 1+a_{1}x+a_{2}x^{2}+.....+a_{p-2}x^{p-2}. $$
The "spectral mapping theorem" tells us that for each eigenvalue $\lambda$ of $m_{\zeta}$, $p(\lambda)$ is an eigenvalue of $p(m_{\zeta}) = m_{p(\zeta)} = m_\alpha$.
Edit: With your latest change: suppose $a_1 \neq 0$. We have $\alpha = 1 + a_1 \zeta = a_1(\zeta + a_1^{-1})$. The determinant of $m_\alpha$ is simply given by $$ \det(m_{\alpha}) = \det(a_1(m_{\zeta} - a_1^{-1})) = a_1^{p-2} \det(m_{\xi} - a_1^{-1}) = a_1^{p-2}\chi_{\zeta}(a_1^{-1}). $$