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]
[23] Jahresber. d. Deutschen Math.-Vereinigung, "Ueber die Theorie der algebraischen Zahlkörper," vol. 4 (1897), Part V.
[24] Math. Annalen, vol. 51 and Nachrichten d. K. Ges. d. Wiss. zu Göttingen, 1898.