The Algebra of the eth Power Residues

The main object of the present paper is to show that the algebra of the Gaussian periods is identical with the algebra of the e^th power residues mod p where p is a prime and e is a divisor of p-1. This algebra is obviously a particular case of the algebra of the e^th power residues mod N where N is any integer. Since the system of residue classes mod N is abstractly identical with a cyclic group of order N, the above algebra may be interpreted to be the algebra of the e^th power residues connected with a cyclic group of order N. It is therefore natural to enquire whether there is a similar algebra connected with any finite Abelian group. We shall however show by means of an example that this is not the case.