Mathematical Problems

Chapter 10: 9. PROOF OF THE MOST GENERAL LAW OF RECIPROCITY IN ANY NUMBER FIELD.

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A systematic collection of fundamental open problems is presented to chart directions for future mathematical research. Topics range across set theory, the consistency of arithmetic, geometry, transformation groups, number theory, algebraic forms, analysis, the calculus of variations, and the mathematical formulation of physical axioms. Challenges highlighted include questions about the continuum, irrationality and transcendence, the distribution and properties of primes, reciprocity laws, solvability of Diophantine equations, uniformization and boundary-value problems, and the finiteness or structure of function systems. Each problem is posed to emphasize its technical difficulty and its capacity to stimulate new methods and broader theoretical development.

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.