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The Reciprocity Law for l-th Power Residues Between a Rational Number and a Number in the Field of l-th Roots of Unity

In: The Theory of Algebraic Number Fields

Author

Listed:
  • David Hilbert

Abstract

Let l be an odd prime number, $$\varsigma = {e^{2\pi i/l}}$$ and k(ζ) the cyclotomic field generated by ζ. Let p be a rational prime number distinct from l and p a prime ideal of k(ζ) dividing p. If p has degree f then, according to Theorem 24, we have for every integer a of k(ζ) not divisible by p the congruence $${\alpha ^{{p^{f - 1}}}} - 1 = 0$$ (mod p).

Suggested Citation

  • David Hilbert, 1998. "The Reciprocity Law for l-th Power Residues Between a Rational Number and a Number in the Field of l-th Roots of Unity," Springer Books, in: The Theory of Algebraic Number Fields, chapter 25, pages 199-205, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03545-0_25
    DOI: 10.1007/978-3-662-03545-0_25
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