Related papers: Undecidability, unit groups, and some totally imag…

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…

We prove that arithmetic is interpretable in any indecomposable polynomial ring (in any set of variables), and in addition we provide an alternative uniform proof of undecidability for all members in this class of rings.

The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of…

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with…

In this paper, we revisit the theory of perfect unary forms over real quadratic fields. Specifically, we deduce an infinite family of real quadratic fields $\mathbb{Q}(\sqrt{d})$ when $d=2$ or $3$ mod $4$, such that there are three classes…

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of $\Q$, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…

The ordered structures of natural, integer, rational and real numbers are studied in this thesis. The theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order…

We show that Morley's theorem on the number of countable models of a countable first-order theory becomes an undecidable statement when extended to second-order logic. More generally, we calculate the number of equivalence classes of…

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…

The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as…

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe-Fehm in a strong way. Along the…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…

While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq…

We realize infinitely many covering groups $2.A_n$ (where $A_n$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works…

In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language…

Let $D_n(x;a)$ and $E_n(x;a)\in\mathbb F_q[x]$ be Dickson polynomials of first and second kind respectively, where $\mathbb F_q$ is a finite field with $q$ elements. In this article we show explicitly the irreducible factors these…

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n = (q+1)/2$. For each $s \in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y): = y^n + (1-y)^n -s$, and we give a necessary and…