algebra and geometry
geometries of matrices. i. generalizations of von staudt's theorem
geometries of matrices. il. arithmetical construction
orthogonal classification of hermitian matrices
geometries of matrices. ii. study of involutions in the geometry of symmetric matrices
geometries of matrices. iii. fundamental theorems in the geometries of symmetric matrices
some “anzahl” theorems for groups of prime power orders
on the automorphisms of the symplectic group over anyfield
on the existence of solutions of certatin equations in a finite field
characters over certain types of rings with applications to the theory of equations in a finite field
on the automorphisms of a sfield
on the number of solutions of some trinomial equations in a finite field
on the nature of the solutions of certain equations in a finite field
some properties of a sfield
on the generators of the symplectic modular group
geometry of symmetric matrices over any field with characteristic other than two
on the multiplicative group of a field
a theorem on matrices over a sfield and its applications
supplement to the paper of dieudonne on the automorphisms of classical groups
automorphisms of the unimodular group
automorphisms of the projective unimodular group
It was first shown in the author's recent investigations on the theory of auto-morphic functions of a matrix-variable that there are three types of geometry playingimportant roles. Besides their applications, the author obtained a great many resultswhich seem to be interesting in themselves.
The main object of the paper is to generalize a theorem due to von Staudt, whichis known as the fundamental theorem of the geometry in the complex domain. Thestatement of the theorem is:
Every topological transformation of the complex plane into itself, which leavesthe relation of harmonic separation invariant, is either a eollineation or an anti-collineation.
Since the fields and groups may be varied, several generalizations of vonStaudt's theorem will be given. The proofs of the theorems have interestingcorollaries.
The paper contains also some fundamental results which will be useful in suc-ceeding papers.