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

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).