Prove that every minor in a matrix is invertible. Let $\zeta_p=e^{\frac{2\pi i}{p}}$ where $p$ is a prime. Let $C=
\begin{pmatrix}
\zeta_p^{0 \times 0} & \zeta_p^{0 \times 1} & \dots & \zeta_p^{0 \times (p-1)} \\
\zeta_p^{1 \times 0} & \zeta_p^{1 \times 1} & \dots & \zeta_p^{1 \times (p-1)} \\
\vdots  &      \vdots &  & \vdots \\
\zeta_p^{(p-1) \times 0} & \zeta_p^{(p-1) \times 1} & \dots & \zeta_p^{(p-1) \times (p-1)}
\end{pmatrix}.$
Prove that every minor in this matrix is invertible.
This matrix can be viewed as the character table of $\mathbb{Z}_p$, which may be helpful.
 A: For the DFT matrix $F_{p}$ of size $p$ where 
\begin{eqnarray*}
(F)_{m,n}=\exp\left(\jmath \frac{2 \pi }{p}  \right)^{(m-1)(n-1)}
\end{eqnarray*}
 is circulant.  The inverse matrix $\left(F^{-1} \right)_{m,n} = \frac{1}{p} \exp\left(-\jmath \frac{2 \pi }{p}  \right)^{(m-1)(n-1)}$.  I don't know if there is a formula for any circulant matrix, but for DFT matrix, the inverse of a submatrix (i.e., any $p-1 \times p-1$ submatrix of $F_{p}$  can be expressed 
in terms of the inverse DFT matrix. 
Let $G_{[u,v]}$ is the submatrix of $F$  by puncturing the row $u$ and column $v$ of the matrix $F$. Then we can establish that,
\begin{eqnarray*}
\left(G_{[u,v]}^{-1} \right)_{m,n} &=& \frac{1}{p} \frac{\exp\left(\jmath \frac{2 \pi}{p} \right)^{m n} -\exp\left(\jmath \frac{2 \pi}{p} \right)^{m u + n v - u v}  }{\exp\left(\jmath \frac{2 \pi}{p} \right)^{m+n-1}}, m,n \in [1,p-1]
\end{eqnarray*}
I don't know how easy it is to prove this generically for any circulant matrix (playing around this a bit for some small circulant matrices beyond DFT, it does seem to have a nice structure). It will be nice if the community can get together to prove this and expand further.  For the DFT matrix, this does hold. 
Here is a matlab code snippet, if you wish to play with. 

% G     --- The G=minor(F,m,n), which is a submatrix of F, where row m and 
%           column n are punctured. Claim is that C=inv(G)
N=4;                % DFT siize
w=exp(1i*2*pi/N);    % DFT nernem
F=dftmtx(N);         % DFT matriix
invF=inv(F);         % iiDFT matriix 
G=F;

P=zeros(size(F));
for ii=1:N
    for jj=1:N
        G=F;
        G(ii,:)=[];
        G(:,jj)=[];
        invG=inv(G);
        C=[];
        for n=1:1:N
            for m=1:N
                C(m,n)=1/N*(w^-(m+n-1).*(w^(n*m)-w^(n*ii+jj*m-ii*jj)));
            end
        end
        C(ii,:)=[];
        C(:,jj)=[];
        C=C.';
        P(ii,jj)=max(sum(abs(invG-C)));
    end
end

MaxErr=max(P(:))>1e-10


A: Edit (04-25) : This (Vandermonde) matrix is well known as the Discrete Fourier Transform (DFT) matrix. 
This issue is in fact established as the Chebotarëv theorem on roots of unity (there are other theorems by the same mathematician). See here for an accessible proof ; here, I found a reference to this excellent text giving (see "Problem" page 5) with a proof using $p$-adic framework. Recent developments here.

Here is separate proof for the case of $(p-1) \times (p-1)$ minors I had given at first.
The DFT matrix $C$ of order $p$ has the property that 
$$C\bar C=pI \tag{1}$$
where the bar means "complex conjugate" (see "The Unitary matrix" in this reference).
Remark : an immediate consequence of (1) is that $|\det(C)|=\sqrt{p}\ne 0$.
Otherwise said 
$$C^{-1}=\frac1p \bar C$$
As $C^{-1}$ is $\dfrac{1}{\det(C)}$ times the transpose of the comatrix of $C$, it follows that the cofactor (and as a consequence, the minor) of entry $C_{mn}$ has the form $k \overline{C_{mn}}$ with $k \ne 0$. 

Edit (04-30) : One can understand the structure of the $\binom{p}{s}^2$  determinants of all the $s \times s$ submatrices of the $n \times n$ matrix $C$ by taking an example in the low-dimensional cases:
$$ p = 5 \ \ \ \text{and} \ \ \ s=3, \ \ \text{with notation : } \ w=e^{2i\pi/p}$$ 
These $10^2$ determinants (some of them listed below, obtained by a Matlab program given at the end) have common properties : 


*

*there common degree in "variable" $w$ is $p-1=4$ and

*they are all divisible by $(w-1)$ 
(I have no proof for these properties). 
$$\begin{array}{rr}    -(w - 1)(2w^3 + 2w^2 + 1),&  (w + 1)(w - 1)^3\\
                 (w - 1)(w^2 + 3w + 1),& (2w^2 + 2w + 1)(w - 1)^2\\
                w(w - 1)(2w^2 + w + 2), &       w(w - 1)(2w^2 + w + 2)\\
                w(w - 1)(w^2 + 3w + 1), &       w(w - 1)(w^2 + 3w + 1)\\
          \cdots & \cdots  \\
                  (w - 1)(w^2 + 3w + 1), &        -(w - 1)(w^3 + 2w + 2)\\
         -(w - 1)(w + 1)(2w^2 + w + 2), &    -(w - 1)(2w^3 + 2w^2 + 1)\\
                 -(w - 1)(2w^2 + w + 2), &     (w^2 + 2w + 1)(w - 1)^2\\
                  (w - 1)(w^3 + 2w + 2), &         (w - 1)(w^2 + 3w + 1)\\
               -(w^2 + 2w + 2)(w - 1)^2, &               (w + 1)(w - 1)^3\end{array}$$
Had we taken $s=4$, we would have seen that all the determinants are even divisible by $(w-1)^2$ ! 
The generic Matlab program :
clear all:
p=5; % modulus
s=3; % size of submatrices (s<p)
syms w; % symbolic variable
[X,Y]=meshgrid(0:(p-1));C=w.^(X.*Y); % will be DFT matrix
li=p*(p-1)*(2*p-1)/6; % upper bound for polynomial's degrees
S=nchoosek(1:p,s); % all combinations with s elements out of 1..p
[nc,~]=size(S); % nc = number of combinations.
k=1;
for n=1:nc;
   I=S(n,:);
   for m=1:nc
      J=S(m,:);
      d=det(C(I,J)); % det. of submatrix defined by index sets I,J
      %e=subs(d,w,exp(i*2*pi/p));plot(e,'pr');
      for q=1:li
         d=subs(d,w^q,w^(mod(q,p))); % property w^p=1
      end;
      D(k)=factor(d);k=k+1;
    end;
end;
D.', % array of all determinants

The different minors of a same category can be represented graphically (uncomment for that the line beginning by e=subs(...)) : here is for example the plot corresponding to the $100$ values above.

Here is the plot corresponding to the case $p=7$ for $s=2$ (i.e. $2 \times 2$ minors). Once again, one finds a perfect symmetry :

And the plot for the  case $p=7$ for $s=5$ (i.e. $5 \times 5 minors) :

