Determinant of circulant matrices

Let $n$ be a positive integer and $J:=(\delta_{i+1,j})_{1\leqslant i,j\leqslant n}$ where $i$ is taken modulo $n$. Meaning

$$J=\begin{pmatrix} 0 & 1 & & & (0)\\ & 0 & 1 & & \\ & & \ddots&\ddots & \\ &(0) & & \ddots& 1\\ 1 & & & & 0 \end{pmatrix}.$$

1. Let $a_0,\dots, a_{n}$ be some complex number and :

$$M(a_0,\dots,a_n)=(a_{j-i})_{1\leqslant i,j\leqslant n}=\begin{pmatrix} a_0 & a_1 & \dots & \dots & a_{n}\\ a_n & a_0 & a_1 & & \vdots \\ \vdots & \ddots & \ddots&\ddots & \vdots \\ \vdots & & \ddots & \ddots& a_1\\ a_1 & \dots & \dots & a_n & a_0 \end{pmatrix}$$

is what is called a circulant matrix.

Write $M$ only as a function of $J$ and its powers.

3. Determine all the eigenvalues of $M$ thanks to 1) and hereafter show that its determinant is given by the formula

$$\mathrm{det}(M(a_0,\dots,a_n))=\prod_{k=1}^{n} P(e^{2ik\pi/n}),$$where $P(X):=\sum_{i=0}^n a_i X^i$.