By “number theory” I mean algebraic number theory, and by Part 1, roughly the years 1900–1940. The treatment is very sketchy; I hope at some later date to cover the rest of the century and to fill in some of the gaps and correct any errors in this hasty account. At the end of the 19th century, Kurt Hensel introduced the idea of the completion kv of a field k with respect to a discrete valuation...

Given a standard set δ of a finite size r, we show that the functor associating to a k-algebra B the set of all reduced Gröbner bases with standard set δ is representable. We show that the representing scheme Hilb′ δ k[x]/k is a locally closed stratum in the punctual Hilbert scheme Hilbrk[x]/k. Moreover, we cover the punctual Hilbert scheme by open affine subschemes attached to all standard set...

In this paper we extend a result of Šemrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms ...