Arithmetic-geometric mean sequences were already studied over the real and complex numbers, and recently, Griffin et al. [‘AGM and jellyfish swarms of elliptic curves’, Amer. Math. Monthly 130(4) (2023), 355–369] considered them over finite fields $\mathbb {F}_q$ for $q \equiv 3 \pmod 4$. We extend the definition of arithmetic-geometric mean sequences over $\mathbb {F}_q$ to $q \equiv 5 \pmod 8$. We explain the connection of these sequences with graphs and explore the properties of the graphs in the case where $q \equiv 5 \pmod 8$.

Let (K, v) be a valued field and $\phi\in K[x]$ be any key polynomial for a residue-transcendental extension w of v to K(x). In this article, using the ϕ-Newton polygon of a polynomial $f\in K[x]$ (with respect to w), we give a lower bound for the degree of an irreducible factor of f. This generalizes the result given in Jakhar and Srinivas (On the irreducible factors of a polynomial II, J. Algebra 556 (2020), 649–655).

We provide a characterization of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to characterize differential largeness in terms of being existentially closed in the differential algebraic Laurent series ring, and we prove that any large field of infinite transcendence degree can be expanded to a differentially large field even under certain prescribed constant fields. As an application, we show that the theory of proper dense pairs of models of a complete and model-complete theory of large fields, is a complete theory. As a further consequence of the expansion result we show that there is no real closed and differential field that has a prime model extension in closed ordered differential fields, unless it is itself a closed ordered differential field.

Motivated by the recent work of Zhi-Wei Sun [‘Problems and results on determinants involving Legendre symbols’, Preprint, arXiv:2405.03626], we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi $ be the unique quadratic multiplicative character of the finite field $\mathbb {F}_q$. If the set $\{s_1,\ldots ,s_{(q-1)/2}\}=\{x^2:\ x\in \mathbb {F}_q\setminus \{0\}\}$, then we prove that

$$ \begin{align*}\det[t+\phi(s_i+s_j)+\phi(s_i-s_j)]_{1\le i,j\le (q-1)/2}=\bigg(\frac{q-1}{2}t-1\bigg)q^{{(q-3)}/{4}}.\end{align*} $$

This confirms a conjecture of Zhi-Wei Sun.

Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup of $\textrm{Gal}(K)$, then ${\mathrm{rank}}(A)\le r+1$. Moreover, if $\mathrm{char}(K)=0$, then ${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of $\textrm{Gal}(K)$.

We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields $\mathbb {F}$ of characteristic zero as a normalised $\mathbb {F}$-linear functional on $\mathbb {F}[\alpha _1, \alpha _2]$ that maps polynomials that evaluate to zero on C to zero and is $\mathrm {SO}(2,\mathbb {F})$-invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers

$$ \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*} $$

an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials $\alpha _1^{2m}\alpha _2^{2n}$.

We prove that the class of all the rings $\mathbb {Z}/m\mathbb {Z}$ for all $m>1$ is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles $\mathbb {A}_{\mathbb {Q}}$ of $\mathbb {Q}$.

We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers.

Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$-algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

We prove that for any prime power $q\notin \{3,4,5\}$, the cubic extension $\mathbb {F}_{q^{3}}$ of the finite field $\mathbb {F}_{q}$ contains a primitive element $\xi $ such that $\xi +\xi ^{-1}$ is also primitive, and $\operatorname {\mathrm {Tr}}_{\mathbb {F}_{q^{3}}/\mathbb {F}_{q}}(\xi )=a$ for any prescribed $a\in \mathbb {F}_{q}$. This completes the proof of a conjecture of Gupta et al. [‘Primitive element pairs with one prescribed trace over a finite field’, Finite Fields Appl. 54 (2018), 1–14] concerning the analogous problem over an extension of arbitrary degree $n\ge 3$.

We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$, where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds, in terms of the minimal polynomials of $\cos ({2\pi }{/d})$. The method of proof is based on earlier work, D. A. Wolfram, [‘Factoring variants of Chebyshev polynomials of the first and second kinds with minimal polynomials of $\cos ({2 \pi }/{d})$’, Amer. Math. Monthly 129 (2022), 172–176] for factoring variants of Chebyshev polynomials of the first and second kinds. We extend this to show that, in general, similar variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$, do not have factors that are minimal polynomials of $\cos ({2\pi }/{d})$.

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$. We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.

Girstmair [‘On an irreducibility criterion of M. Ram Murty’, Amer. Math. Monthly 112(3) (2005), 269–270] gave a generalisation of Ram Murty’s irreducibility criterion. We further generalise these criteria.

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$, $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$, we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

For $c \in \mathbb {Q}$, consider the quadratic polynomial map $\varphi _c(z)=z^2-c$. Flynn, Poonen, and Schaefer conjectured in 1997 that no rational cycle of $\varphi _c$ under iteration has length more than $3$. Here, we discuss this conjecture using arithmetic and combinatorial means, leading to three main results. First, we show that if $\varphi _c$ admits a rational cycle of length $n \ge 3$, then the denominator of c must be divisible by $16$. We then provide an upper bound on the number of periodic rational points of $\varphi _c$ in terms of the number s of distinct prime factors of the denominator of c. Finally, we show that the Flynn–Poonen–Schaefer conjecture holds for $\varphi _c$ if $s \le 2$, i.e., if the denominator of c has at most two distinct prime factors.

We analyse the behaviour of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field $\mathbb{F}_{q}$ of q elements when the highest degree polynomial g is fixed. Considering all the elements f of fixed degree, we establish asymptotically optimal bounds in terms of q for the number of elements f that are relatively prime with g and for the average degree of $\gcd(g,f)$. We also exhibit asymptotically optimal bounds for the average-case complexity of the Euclidean algorithm applied to pairs (g,f) as above.

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.

We introduceand study anatural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. Weshow that global fields belong to this class, and in particular obtain a number of new existential (or diophantine) predicates over global fields.

Let n be a positive integer and let $\mathbb{F} _{q^n}$ be the finite field with $q^n$ elements, where q is a prime power. We introduce a natural action of the projective semilinear group ${\mathrm{P}\Gamma\mathrm{L}} (2, q^n)={\mathrm{PGL}} (2, q^n)\rtimes {\mathrm{Gal}} ({\mathbb F_{q^n}} /\mathbb{F} _q)$ on the set of monic irreducible polynomials over the finite field $\mathbb{F} _{q^n}$. Our main results provide information on the characterisation and number of fixed points.

We answer a question posed by Mordell in 1953, in the case of repeated radical extensions, and find necessary and sufficient conditions for $[F[\sqrt [m_1]{N_1},\dots ,\sqrt [m_\ell ]{N_\ell }]:F]=m_1\cdots m_\ell $, where F is an arbitrary field of characteristic not dividing any $m_i$.

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $xy$, where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.