Related papers: Artin-Schreier extensions in NIP and simple fields
We defined (n+1)-ary derivations of $n$-ary algebras. We described (n+1)-derivations of simple and semisimple finite-dimensional Filippov algebras over algebraically closed field zero characteristic. We constructed new examples of…
We construct and study fields F with the property that F has infinitely many extensions of some fixed degree, but E*/(E*)^n is finite for every finite extension E of F and every n>0.
Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where…
In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real…
We give a simple derivation of the formula for the number of normal elements in an extension of finite fields. Our proof is based on the fact that units in the Galois group ring of a field extension act simply transitively on normal…
We prove that no infinite field is interpretable in the first-order theory of nonabelian free groups. We also obtain a characterization of Abelian groups interpretable in this theory.
Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p>0$ and $B \subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number…
We define an additive version of the Bloch group of a field, together with an additive dilogarithm and an Artin-Schreier realization.
We characterize extensions of commutative rings $R\subset S$ such that $R\subset T$ is minimal for each $R$-subalgebra $T$ of $S$ with $T\neq R,S$. This property is equivalent to $R\subset S$ has length 2. Such extensions are either…
We study the repetition of patches in self-affine tilings in R^d. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling…
We prove for a large class of fields $F$ that every proper finite extension of $F_{pyth}$, the pythagorean closure of $F$, is not a pythagorean field. This class of fields contains number fields and fields $F$ that are finitely generated of…
We propose a slight weakening of the definitions of Artin monoids and Coxeter monoids. We study one `infinite series' in detail.
We study just infinite algebras which remain so upon extension of scalars by arbitrary field extensions. Such rings are called stably just infinite. We show that just infinite rings over algebraically closed fields are stably just infinite…
Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension $n$ to algebras and categories to include Auslander algebras and a graded…
In this paper we intend to show that certain integers do not occur as the norms of principal ideals in a family of cubic fields studied by Cohn, Shanks, and Ennola. These results will simplify the construction of certain unramified…
We introduce a new invariant of fields that refines their real spectrum and is related to their absolute Galois group: the Artin-Schreier quandle. For formally real number fields, it is freely generated in its variety by a Cantor space of…
We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…
The ring of integers and the discriminant are determined for number fields which are simple radical extensions.
We give general conditions under which classes of valued fields have NIPn transfer and generalize the Anscombe-Jahnke classification of NIP henselian valued fields to NIPn henselian valued fields.
Artin groups of finite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of finite type, we…