In Paper A we classify plethories over a field of characteristic zero. All plethories over characteristic zero fields are ``linear", in the sense that they are free plethories on a bialgebra. For the proof of this classification we need some facts from the theory of ring schemes where we extend previously known results. We also give a classification of plethories with trivial Verschiebung over a perfect field k of characteristic p>0.

In Paper B we study tensor products of abelian affine group schemes over a perfect field k. We first prove that the tensor product G_1 \otimes G_2 of two abelian affine group schemes G_1,G_2 over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 \otimes G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. In characteristic zero the unipotent part of G_1 \otimes G_2 is the group scheme whose primitive elements are P(G_1) \otimes P(G_2). In positive characteristic, we give a formula for the tensor product in terms of Dieudonné theory. 

In Paper C we use ideas from homotopy theory to define new obstructions to solutions of embedding problems and compute the étale cohomology ring of the ring of integers of a totally imaginary number field with coefficients in Z/2Z. As an application of the obstruction-theoretical machinery, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2,q^2)), for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. 

In Paper D we compute the étale cohomology ring of an arbitrary number field with coefficients in Z/nZ for n an arbitrary positive integer. This generalizes the computation in Paper C. As an application, we give a formula for an invariant defined by Minhyong Kim.

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work. 

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work.

We give formulas for 3-fold Massey products in the étale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of p-rank two possessing an infinite p-class field tower, where p is an odd prime. We also disprove McLeman's (3,3)-conjecture.

We compute the cohomology ring H*(U,ℤ/nℤ) for U=X∖S where X is the spectrum of the ring of integers of a number field K and S is a finite set of finite primes.