For any field of numbers the law of reciprocity is to be proved for the residues of the th power, when denotes an odd prime, and further when is a power of or a power of an odd prime.

The law, as well as the means essential to its proof, will, I believe, result by suitably generalizing the theory of the field of the th roots of unity,[23] developed by me, and my theory of relative quadratic fields.[24]