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The l-th Power Reciprocity Law in Regular Cyclotomic Fields

In: The Theory of Algebraic Number Fields

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  • David Hilbert

Abstract

The theory of Kummer fields which we have developed thus far gives us the resources required for the proof of certain fundamental laws concerning l-th power residues in regular cyclotomic fields which correspond to the quadratic reciprocity law in the domain of rational numbers and include as a special case the Eisenstein reciprocity law (Theorem 140) between an arbitrary number in k(ζ) and a rational number which we developed in Sect. 115. In order to be able to state these laws for l-th power residues in their simplest form we generalise the symbol {μ/m} defined in Sect. 113 and Sect. 127 as follows.

Suggested Citation

  • David Hilbert, 1998. "The l-th Power Reciprocity Law in Regular Cyclotomic Fields," Springer Books, in: The Theory of Algebraic Number Fields, chapter 33, pages 289-304, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03545-0_33
    DOI: 10.1007/978-3-662-03545-0_33
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