What actually complex roots are?

Question

I solved the equation $$x^7 - 5=0$$with the help of MATLAB and found seven roots as expected but the thing which wonders me is that all the complex roots have same absolute value aand that is equal to the real root among those seven, what complex roots actually show? Is my hypothesis "All polynomials must have at least one real root" is true? Also calculator gives the value equal to the absolute value of the roots we I solve $\sqrt[7]{5}$ Here is the MATLAB code

x=[1 0 0 0 0 0 0 -5];

b=roots(x)

b =

-1.1339 + 0.5460i

-1.1339 - 0.5460i

-0.2800 + 1.2269i

-0.2800 - 1.2269i

0.7847 + 0.9839i

0.7847 - 0.9839i

1.2585 + 0.0000i

abs(b)

ans =

1.2585

1.2585

1.2585

1.2585

1.2585

1.2585

1.2585

Answer 1

It is well-known that all $n$-th roots of a complex number $z_0$ differ by a factor which an $n$-th root of unity; in other words: once you have a solution $\zeta$ of the equation $z^n=z_0$, all other solutions have the form $$a=\zeta \omega,\quad\text{where}\enspace \omega^n=1.$$