Let $K$ be a global field, $\cal{V}$ an infinite proper subset of the set of all primes of $K$, and $\cal{S}$ a finite subset of $\cal{V}$. Denote the maximal Galois extension of $K$ in which each $\mathfrak{p}\in\cal{S}$ totally splits by $K_{tot,S}$. Let $M$ be an algebraic extension of $K$. Let $\cal{V}_M$ (resp. $\cal{S}_M$) be the set of primes of $M$ which lie over primes in $\cal{V}$ (resp. $\cal{S}$). For each $\mathfrak{q}\in\cal{V}_M$ let $\hat{cal{O}}_{M,\mathfrak{q}}=\{x\in\hat{M}_{\mathfrak{q}}|\;|x|_{\mathfrak{q}}\le 1\}$, where $\hat{M}_{\mathfrak{q}}$ is a completion of $M$ at $\mathfrak{q}$, and let $\cal{O}_{M,\cal{V}}=\{x\in M|\;|x|_{\mathfrak{q}}\le 1\ \hbox{{\rm for each}} \mathfrak{q}\in\cal{V}_M\}$.

For $\bf{\sigma}=(\sigma_1,\dots,\sigma_e)\in Gal(K)^e$, let $K_s(\bf{\sigma})=\{x\in K_s |\; \sigma_i(x)=x,\,i=1,\dots,e\}$. Then, for almost all $\bf{\sigma}\in Gal(K)^e$ (with respect to the Haar measure), the field $M=K_s(\bf{\sigma})\cap K_{tot,S}$ satisfies the following local global principle: Let $V\subseteq\mathbb{A}^n$ be an affine absolutely irreducible variety defined over $M$. Suppose that there exist $\bf{x}_{\mathfrak{q}}\in V(\hat{\cal{O}}_{M,\mathfrak{q}})$ for each $\mathfrak{q}\in\cal{V}_M\setminus\cal{S}_M$ and $\bf{x}_{\mathfrak{q}}\in V_{\rm simp}(\hat{\cal{O}}_{M,\mathfrak{q}})$ for each $\mathfrak{q}\in\cal{S}_M$ such that $|x_{i,\mathfrak{q}}|_{\mathfrak{q}} \lt 1$, $i=1,\dots,n$, for each archimedean prime $\mathfrak{q}\in\cal{V}_M$. Then $V(\cal{O}_{M,\cal{V}})\ne\emptyset$.

We present the main results of the theory of non-unique factorizations as far as they deal with algebraic integers. We specify the philosophy that the class group of an algebraic number field measures to what extent its ring of integers fails to have unique factorization. On the other hand, if the ring of integers fails to have unique factorization then (in a sense to be made precise) almost all integers have many distinct factorizations, but also almost all integers have a clearly arranged set of factorizations.

It is proved that for every positive $B$ there exist real numbers $0=a_0<a_1<\ldots <a_N=1$ and $ \max_{1\leq j\leq N} (a_{j-1}/a_j)\leq\theta<1$ such that $$\limsup_{x\to\infty} \frac{1}{\sqrt{x}}\sum_{j=1}^N\sum_{\theta a_j x<n\leq a_j x} \mu(n) \geq B$$ and $$\liminf_{x\to\infty}\frac{1}{\sqrt{x}}\sum_{j=1}^N\sum_{\theta a_j x<n\leq a_j x} \mu(n)\leq -B,$$ where $\mu(n)$ denotes the Möbius function.

We prove that the expressible sets of geometric sequences are Borel measurable and give lower and upper bounds for their Lebesgue measure.

We give an efficient algorithm for solving resultant form equations over global function fields. This is the first time that such equations are reduced to unit equations in two variables and all solutions are determined.

We state an effective version of the $abc$-conjecture, and assuming it we describe all solutions of some diophantine equations. Based on this conjecture we give an effective algorithm for computing an infinite set of primes which are not Wieferich primes.

All complex, pure quartic fields with maximal orders generated by their units are determined. Furthermore, a quantitative version of the unit sum number problem is considered.

Let $a$ and $f$ be coprime positive integers. Let $g$ be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes $p$ such that $p\equiv a({\rm mod~}f)$ and $g$ is a primitive root modulo $p$ has a natural density. In this note this density is explicitly evaluated with an Euler product as result. This extends a classical result of Hooley (1967) on Artin's primitive root conjecture. Various application are given, for example the integers $g$ and $f$ such that the set of primes $p$ such that $g$ is a primitive root modulo $p$ is equidistributed modulo $f$ is determined (on GRM).

For the cyclotomic extension $F(\mu_{\infty})=\bigcup_{m\geq 1} F(\mu_m)$ of a number field $F,$ we prove that the reduction map $K_{2n{+}1}(F(\mu_{\infty}))\longrightarrow K_{2n{+}1}(\kappa_{{\tilde v}}),$ when restricted to nontorsion elements, is surjective. Here $\kappa_{{\tilde v}}$ denotes the residue field at a prime ${\tilde v}$ of $F(\mu_{\infty}).$

Let $H$ be a Krull monoid such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). For $k \in \mathbb{N}$ let $\mathcal{V}_k (H)$ denote the set of all $m \in \mathbb{N}$ \ with the following property{\rm \,:} There exist atoms (irreducible elements) \ $u_1, \ldots, u_k, v_1, \ldots, v_m \in H$ with $u_1 \cdot \ldots \cdot u_k = v_1 \cdot \ldots \cdot v_m$. We show that the sets $\mathcal{V}_k (H)$ are intervals for all $k \in \mathbb{N}$. This solves Problem 37 in [4].

This paper surveys on some recent results concerning certain finiteness properties for subfields $K$ of $\overline{\mathbb{Q}}$: first, the so-called \textit{Northcott property} of finiteness of elements in $K$ of bounded Weil height and then the \textit{Property (P)} of finiteness of possible subsets of $K$ sent onto themselves by some polynomial of degree $>1$. The first was established by Northcott for the union of the fields of given degree over $\mathbb{Q}$; the second one was introduced by Narkiewicz; it is also related to preperiodic points for polynomial maps. It is known that the first implies the second, so they both hold for number fields. As to fields of infinite degree over $\mathbb{Q}$, we shall see some criteria for the first property, and hence for the second, but we shall also see that the second property does not imply the first. Some of these constructions provide answers, both in the positive and in the negative as the case may be, to questions explicitly formulated by Narkiewicz.