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

David Hilbert

Chapter 25 in The Theory of Algebraic Number Fields, 1998, pp 199-205 from Springer

Abstract: 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).

Date: 1998
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DOI: 10.1007/978-3-662-03545-0_25

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