Stickelberger's thesis and several later papers streamline and complete earlier investigations of various authors, in a direct and elegant way.

Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and Darboux.

Augmented with the contemporaneous work of Frobenius, it set the theory of elementary divisors upon a rigorous foundation.

An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the classification of finitely generated abelian groups and sketched the relation with the theory of modules that had just been developed by Dedekind.

This generalized earlier work of Jacobi and Kummer and was later used by Hilbert in his formulation of the reciprocity laws in algebraic number fields.