ABSTRACT
In Chapter 3, we started with some elementary facts concerning power residues, and then we proceeded with quadratic residues, quadratic reciprocity and its consequences. To obtain a similar theory including reciprocity laws for higher power residues, one has to leave the rational number field. Already Gauss noticed that in order to formulate a reciprocity law for cubic or biquadratic residues it is necessary to use complex integers, but it was only Eisenstein who gave complete proofs for the biquadratic and cubic reciprocity laws using the cyclotomic fields of fourth and third roots of unity. Moreover, he succeeded in proving a special case of the general reciprocity law for l-th power residues, where l is an odd prime. Only class field theory, as developed by Takagi, Hasse and Artin, provided a full insight into higher reciprocity laws. Here we must limit ourselves to these sketchy remarks, for a full history of the subject and more details we refer to Lemmermeyer’s book [67].
