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A formula for power residue symbols is deduced, which can be treated as an analog of the Abelian integral theorem for number fields. §0. Introduction It was Leopold Kronecker who first understood that there is a deep relationship between algebraic numbers and algebraic functions. He stated that the prime ideals in the fields of algebraic functions play the same role as the points of Riemann surfaces: the ramification points of a Riemann surface correspond to prime divisors of the discriminant of a number field, etc. The theory of Abelian integrals is the most studied part of the theory of algebraic functions. David Hilbert was apparently the first who initiated an analog of this theory in the fields of algebraic numbers. In particular, he noticed that his reciprocity law for the product of norm residue symbols (Hilbert symbols),

The classical reciprocity law for power residues as an analog of the Abelian integral theorem