Complex roots of unity
School pupils familiar with complex numbers and DeMoivre's theorem know examples of algebraic numbers. For let
(where
n
is any natural number), then
=
= 1
and thus
is a solution of the polynomial equation
.
In fact, the
n
solutions of the equation
are
(=1).
For example, when
, the equation
has 6 (algebraic number) solutions:

one of which will be of degree 1

another of which will also be of degree 1

two of which will be of degree 2

and another two of which will also be of degree 2
>
n := 6:
factor(x^n  1);
>
whereas, for example, when
, something quite different will happen (and I will not elaborate...):
>
n := 7:
factor(x^n  1);
>