9. PROOF OF THE MOST GENERAL LAW OF RECIPROCITY IN ANY NUMBER FIELD.

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]