Hilbert’s Reciprocity Law

Abstract

The Hilbert Reciprocity Law states that

$$\prod\limits_p {\left( {\frac{{\alpha ,\beta }}{p}} \right)} = 1.$$

The major portion of this chapter is devoted to the proof of this formula for algebraic number fields. The formula is actually true over any global field, but we shall not go into the function theoretic case here. The Hilbert Reciprocity Law gives a reciprocity law for Hasse symbols, namely

$$\prod\limits_p {{S_p}V} = 1,$$

and this can be regarded as a dependence relation among the invariants of the quadratic space V. We shall investigate the full extent of this dependence in § 72.