What are the eigenvalues of the 8 point DFT matrix? I  know that the eigenvalues for 4 point DFT matrix can be found from $F_4^4=I$. Is this also valid for 8, 16 and higher orders? For example with  8 points, will it be $F_8^8=I$ ? If not, how  can I can compute them?
The eigenvalues belong to the same set of quartic roots of unity verifying $\lambda^4=1$, whatever the order of the DFT. 

For more details on their multiplicity, you can read: 
Eigenvectors and Functions of the Discrete Fourier Transform, 1982, Dickinson and Steiglitz (online).