De Moivre's theorem is (cosy+isiny)^n=cos(ny)+isin(ny). If we put y=2(pi)/n and n=7, we get:
(cos(2(pi)/7)+isin(2(pi)/7))^7=cos(2(pi))=1. If x=cos(2(pi)/7)+isin(2(pi)/7)) then x^7=1.
We can also write (cos(2(pi)/7)+isin(2(pi)/7)^p=cos(2p(pi)/7)+isin(2p(pi)/7) where p is 0 to 6.
Only for p=0 is the right-hand side real: (...)^0=1=cos(0)+isin(0).
For other values of p there is always an imaginary part because cos(2(pi)/7) and sin(2(pi)/7) can be evaluated as 0.6235 and 0.7818, making the right-hand side 0.6235+0.7818i which is complex. If the complex plane is represented by the usual x-y plane and y is the imaginary part and x the real part, the 7th root of 1 is represented by the vertices of a regular heptagon, and the only real value (y=0) is the point (1,0), implying 1 is the only real root. (Compare this with the 4th root (2(pi)/4=(pi)/2 or 90 degrees) where there are two real roots at the vertices of a square at (-1,0) and (1,0) and two imaginary at (0,1) and (0,-1).)
x^7-1 factors to: (x-1)(x^6+x^5+x^4+x^3+x^2+x+1), and the larger expression has only complex roots.