§ XI

Abstract

Now we will consider equations of the form x ^m= a, in a field K (or occasionally in a ring); the case a=1 has been discussed in § X. As the case a=0 is trivial, we assume a≠0. If then, in the field K, x is a solution of x^m=a, an element x’ of K is also a solution if and only if (x’/x) ^m =1. Therefore, if x^m=a has a solution in K, it has as many solutions as K contains m ^th roots of unity, i.e. roots of X ^m -1.