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Let $\xi_{n}=e^{\frac{2\pi i}{n}}$. I would like to use cramer's rule to solve:

$\left(\begin{array}{ccccc} 1 & 1 & 1 & \cdots & 1\\ 1 & \xi_{n} & \xi_{n}^{2} & \cdots & \xi_{n}^{n-1}\\ 1 & \xi_{n}^{2} & \xi_{n}^{4} & \cdots & \xi_{n}^{2\left(n-1\right)}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & \xi_{n}^{n-1} & \xi_{n}^{2\left(n-1\right)} & \cdots & \xi_{n}^{\left(n-1\right)^{2}} \end{array}\right)\left(\begin{array}{c} C_{0}\\ C_{1}\\ C_{2}\\ \vdots\\ C_{n-1} \end{array}\right)=\left(\begin{array}{c} c_{0}\\ c_{1}\\ c_{2}\\ \vdots\\ c_{n-1} \end{array}\right) $

That is, to find equations for the big Cs in terms of the little cs. But I fear getting lost in the computations. The matrix here is pretty enough that it makes me suspect that someone has done this computation before. Any help would be much appreciated!

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    If m ≤ n, then the matrix V has maximum rank (m) if and only if all αi are distinct. A square Vandermonde matrix is thus invertible if and only if the αi are distinct; an explicit formula for the inverse is known.[2][3][4] https://en.wikipedia.org/wiki/Vandermonde_matrix – 2017-02-08
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    My thought too. – 2017-02-08
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    I know about the vandermonde determinant. The question is, how is the formula for the determinant going to be affected when you use cramer's rule to substitute in the constant vector [c_0, c_1,...,c_(n-1)] in the mth column? Having explicit formulae for this would be ideal. – 2017-02-08

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