Roots of Unity

The simplest field extensions of a field F are those in which we adjoin an nth root of an element c ∈F. To investigate these, it will be valuable for us to consider roots of unity. After all, if α n = c, then the other nth roots of c are of the form ωα, where ω is some nth root of unity, and so it will be relevant whether or not F contains such roots of unity. Note that R, for example, contains the square roots of unity, namely, 1 and — 1, but it contains no higher roots of unity (other than 1) because they are all complex.