Related papers: Number Systems with Simplicity Hierarchies II
The aim of this book is to introduce the reader to the beauty of Algebra, through a journey from the natural numbers to prime fields and finite fields, with some detours. Many books are devoted to the construction of these fields from the…
Primary superfields for a two dimensional Euclidean superconformal field theory are constructed as sections of a sheaf over a graded Riemann sphere. The construction is then applied to the N=3 Neveu-Schwarz case. Various quantities in the…
A new description of free massless superfields of arbitrary superspin $Y$ ($Y>1/2$) is proposed. Following the first-order philosophy, we relax some of the properties (reality, gauge redundancy) of the unconstrained higher spin…
A first-order theory is Noetherian with respect to the collection of formulae $\mathcal{F}$ if every definable set is a Boolean combination of instances of formulae in $\mathcal{F}$ and the topology whose subbasis of closed sets is the…
A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's…
Using the recent advancements in the structure of algebraic groups over imperfect fields, we propose a generalization of Serre's Conjecture I and of results that revolve around it. In particular, we prove that the first Galois cohomology…
We continue the theory of $\tT$-systems from the work of the second author, describing both ground systems and module systems over a ground system (paralleling the theory of modules over an algebra). The theory, summarized categorically at…
A $\Lambda$-tree is a $\Lambda$-metric space satisfying three axioms (1), (2) and (3). We give a characterization of those ordered abelian groups $\Lambda$ for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for…
It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear…
Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_*(FG)$ denote the…
By fully describing the lattice of subfields of some towers of number fields built by iterating square roots, we obtain infinitely many fields, each of them either contradicts Julia Robinson's problem (obtaining a JR-number $4$ which is not…
When reduced from $10$ to $10-d$ dimensions tree-level string theory exhibits an $O(d,d)$ symmetry. This symmetry, which is closely related to T-duality, appears only after certain field redefinitions. We find a simple form for a subset of…
We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global…
We are interested in classical and logarithmic imaginary classes of abelian number fields in connection with Iwasawa theory. For any given odd prime ${\ell}$ and any imaginary abelian number field K, we compute the isotypic components of…
In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the…
We observe that the definition of Shelah's classical $\mathrm{NSOP}_{n}$ hierarchy for first-order theories, for integers $n \geq 3$, can be restated so that it extends to the case where $n$ is replaced with any real number $r \geq 3$.…
Silverman showed that, assuming the $abc$ conjecture, there are $\gg \log x$ non-Wieferich primes base $a$ less than $x$ \cite{silverman}, for all non-zero $a$. This inspired Graves and Murty \cite{Graves}, Chen and Ding \cite{Chen1}…
If the generalized dynamics of K field theories (i.e., field theories with a non-standard kinetic term) is taken into account, then the possibility of so-called twin-like models opens up, that is, of different field theories which share the…
A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order…
A. Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian…