Modular concomitant scales, with a fundamental system of formal covariants, modulo 3, of the binary quadratic

Abstract

which is subject to the transformations of the total linear group C, modulo p (a prime number), a theory which has been developed only to the extent indicated in the summary below.t Examples of such invariantive functions have been constructed by A. Hurwitz, Miss Sanderson, Dickson, and the present author, and it has been proved that the totality of pure invariants of the type forms a finite system when m * 0 (mod p). A variety of construction methods have been invented and certain particular complete systems of seminvariants and invariants derived. The present writer was the first to publish a fundamental system of formal covariants, that of the binary cubic for the modulus 2. The developments which follow are devoted to processes for the generation of complete systems of covariants and, in particular, give an extension of the principle of the modular covariant scale which was employed in the paper, quoted as IV above, on the system of the cubic modulo 2. In Section 6 the theory of concomitant scales is applied in determining a fundamental system of covariants modulo 3 of the binary quadratic, an interesting system composed of 18 invariants and covariants.