Stickelberger's obituary lists the total of 14 publications: his thesis (in Latin), 8 further papers that he authored which appeared during his lifetime, 4 joint papers with
Georg Frobenius and a posthumously published paper written circa 1915. Despite this modest output, he is characterized there as "one of the sharpest among the pupils of Weierstrass" and a "mathematician of high rank". Stickelberger's thesis and several later papers streamline and complete earlier investigations of various authors, in a direct and elegant way.
Linear algebra 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.
Number theory Three joint papers with Frobenius deal with the theory of
elliptic functions. Today Stickelberger's name is most closely associated with his 1890 paper that established the
Stickelberger relation for cyclotomic
Gaussian sums. 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. The Stickelberger relation also yields information about the structure of the
class group of a
cyclotomic field as a module over its abelian
Galois group (cf
Iwasawa theory). S von Elementen der Galoisgruppe G = \{\sigma_a | a \in (\mathbb Z / p\mathbb Z)^*\} = \{\sigma_a | a \in (1, \ldots, (p-1))\} des
Kreisteilungskörpers \mathbb{Q}(\zeta_p), die auf ein beliebiges
Ideal I im
Ring der ganzen Zahlen \mathbb{Z}(\zeta_p) des Kreisteilungskörpers angewandt ein Hauptideal ergeben. S „annihiliert“ das Ideal I, es ist ein Annihilitor der Idealklassengruppe von \mathbb{Q}(\zeta_p). Nach dem Satz von Stickelberger ist S ein Vielfaches des Stickelberger-Elements :\theta = \frac 1 p \sum_{a \in (\mathbb Z / p\mathbb Z)^*} a \sigma_a^{-1}. aus \mathbb{Q}[G], das in \mathbb{Z}[G] liegt (das heißt, es gibt ein \beta \in \mathbb{Z}[G], so dass S = \beta\theta \in \mathbb{Z}[G]). --> ==References==