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Quadratic Reciprocity in Number Fields

Eisenstein [22] realized that the reciprocity laws for quadratic and cubic residues can be formulated in a way that allows the immediate generalization to arbitrary powers and number fields: his observation was that the inversion factor i(α, β) = (αβ )( β α ) −1 for p-th power residue symbols in Q(ζp) and odd primes p only depends on α, β mod (1− ζ) for some integer m. In this section we show that Eisenstein’s version for p = 2 over Q is equivalent to the quadratic reciprocity law; then we will prove its generalization for quadratic number fields. In the next chapter we will then show how to extract the reciprocity law for p-th power residues from the above conjecture. For coprime odd integers a, b ∈ Z, define a symbol [a, b] with values in {0, 1} by (ab )( b a ) = (−1) . Let us make the following assumption: Hypothesis (H). There exists an integer σ ∈ N such that [a, b] = [a′, b′] for all natural numbers a, b, a′, b′ ∈ N with (a, b) = (a′, b′) = 1, and a ≡ a′, b ≡ b′ mod 2. We could also allow integers a, b, a′, b′ ∈ Z, but then we would have to add the condition that aa′ > 0 and bb′ > 0. In that case, (H) may also be assumed to hold for b = −1; note that ( a −1 ) = ( a 1 ) = 1, since by definition of the Jacobi symbol (a1 ) = ∏ p|1( a p ), and this product is empty. In order to better understand what Hypothesis (H) is all about we first derive the following result (which will not be used in the sequel):