Related papers: Integers for Simple Radical Extensions

In this paper, we investigate generalizations of the Mahler-Popkens complexity of integers. Specifically, we generalize to $k$-th roots of unity, polynomials over the naturals, and the integers mod $m$. In cyclotomic rings, we establish…

A general simplicity problem in category theory is proposed. A particular example, the simplest choice of generators of an algebra is specified and illustrated by an example.

We determine the average number of distinct subsequences in a random binary string, and derive an estimate for the average number of distinct subsequences of a particular length.

A positive integer $n$ is called practical if all integers between $1$ and $n$ can be written as a sum of distinct divisors of $n$. We give an asymptotic estimate for the number of integers $\le x$ which have a practical divisor $\ge y$.

A compound determinant identity for minors of rectangular matrices is established. As an application, we derive Vandermonde type determinant formulae for classical group characters.

In this paper, we describe restricted one-dimensional central extensions of all finite dimensional simple restricted Lie algebras defined over fields of characteristic $p\ge 5$.

After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with…

We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the…

We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.

Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric…

We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.

We consider finite dimensional representations of the dihedral group $D_{2p}$ over an algebraically closed field of characteristic two where $p$ is an odd integer and study the degrees of generating and separating polynomials in the…

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…

In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \leq 5$. Further we give some connections…

Let $R$ be a ring and $P$ a prime ideal of $R.$ In this paper, we establish some commutativity criteria for the factor ring $R/P$ in terms of derivations of $R$ satisfying some algebraic identities involving a new kind of involution in…

A criterion for determining exactly when an order of a maximal subfield of a central simple algebra over a number field can be embedded into an order of this algebra is given. Various previous results have been generalized and recovered by…

For a prime number $p$, an integer $e\geq 2$ and a field $F$ containing a primitive $p^e$-th root of unity, the index of central simple $F$-algebras of exponent $p^e$ is bounded in terms of the $p$-symbol length of $F$. For a nonreal field…

In 2007, Jarden and Narkiewicz raised the following question: Is it true that each algebraic number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? In this article, we answer the…

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the…

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal…