Let K be an infinite field. If $\alpha $ and $\beta $ are algebraic and separable elements over K, then by the primitive element theorem, it is well known that $\alpha +u\beta $ is a primitive element for $K(\alpha , \beta )$ for all but finitely many elements $u\in K$. If we let

$$ \begin{align*}\xi_K(\alpha, \beta) = \{u\in K : K(\alpha, \beta) \ne K(\alpha+u\beta)\}\end{align*} $$

be the exceptional set, then by the primitive element theorem, $|\xi _K(\alpha , \beta )| < \infty $. Dubickas [‘An effective version of the primitive element theorem’, Indian J. Pure Appl. Math. 53(3) (2022), 720–726] estimated the size of this set when $K = \mathbb {Q}$. We take K to be a finite extension over $\mathbb {Q}$ or $\mathbb {Q}_p$, the field of p-adic numbers for some prime p, and estimate the size of the exceptional set.

We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree d, over any global field. In particular, we focus on the affine hypersurface situation by relaxing the condition on the top degree homogeneous part of the polynomial describing the affine hypersurface, while sharpening the dependence on the degree in the bounds compared to previous results. We formulate a conjecture about plane curves which provides a conjectural approach to the uniform degree $3$ case (the only remaining open case). For induction on dimension, we develop a higher-dimensional effective version of Hilbert’s irreducibility theorem, which is of independent interest.

We prove motivic versions of mass formulas by Krasner, Serre, and Bhargava concerning (weighted) counts of extensions of local fields.

In this article, we show that any $\mathbb {A}^2$-fibration over a discrete valuation ring which is also an $\mathbb {A}^2$-form is necessarily a polynomial ring. Further, we show that separable $\mathbb {A}^2$-forms over principal ideal domains are trivial.

This note provides an alternative proof of a theorem by Li et al. [‘On the primitivity of some trinomials over finite fields’, Adv. Math. (China) 44(3) (2015), 387–393] regarding the nonprimitivity of the trinomial $x^{n}+ax+b$ over $\mathbb {F}_{q^{m}}$ under the condition $a^{n}b^{1-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. We extend this result to the trinomial $x^{n}+a^{k}x^{k}+b^{k}$, showing its nonprimitivity over $\mathbb {F}_{q^{m}}$ when $ a^{n}b^{k-n}\in \mathbb {F}_{q^{u}}^{\ast }$ for some positive integer $u<m$. While the existing proof relies on the theory of linear recurrences over finite fields, our approach is short and self-contained, requiring no prior knowledge of this area.

We study universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example, we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

Dedicated to the memory of Alexander Prestel (1941–2024)

We consider a finite-dimensional vector space $W\subset K^E$ over a field K and a set E. We show that the set $\mathcal {C}(W)\subset 2^E$ of minimal supports of W are the circuits of a matroid on E. When the cardinality of K is large (compared to that of E), then the family of supports of W is a matroid. Afterwards we apply these results to tropical differential algebraic geometry (tdag), studying the set of supports of spaces of formal power series solutions $\text {Sol}(\Sigma )$ of systems of linear differential equations (ldes) $\Sigma$ in variables $x_1,\ldots ,x_n$ having coefficients in . If $\Sigma $ is of differential type zero, then the set $\mathcal {C}(Sol(\Sigma ))\subset (2^{\mathbb {N}^{m}})^n$ of minimal supports defines a matroid on $E=[n]\times \mathbb {N}^{m}$, and if the cardinality of K is large enough, then the set of supports is also a matroid on E. By applying the fundamental theorem of tdag (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system U of tropical ldes is a matroid. We give a counterexample to the fttdag for systems $\Sigma $ of ldes over countable fields for which is not a matroid.

We study infinite groups interpretable in power bounded T-convex, V-minimal or p-adically closed fields. We show that if G is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups) then, up to a finite index subgroup, it is definably isogenous to a group $G_1\times G_2$, where $G_1$ is a K-linear group and $G_2$ is a $\mathbf {k}$-linear group. The analysis is carried out by studying the interaction of G with four distinguished sorts: the valued field K, the residue field $\mathbf {k}$, the value group $\Gamma $, and the closed $0$-balls $K/\mathcal {O}$.

We show that each local field $\mathbb{F}_q(\!(t)\!)$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most 1 by (certain small quotients of) its absolute Galois group together with natural axioms concerning the p-torsion of its Brauer group. This complements previous work by Efrat and Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic p.

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in q. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed).

The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma $ which is locally infinitely contracting and fails to be onto. Namely, we prove the existence of a model complete theory $\widetilde {\mathrm {VFE}}$ amalgamating the theories $\mathrm {SCFE}$ and $\widetilde {\mathrm {VFA}}$ introduced in [5] and [11], respectively. In characteristic zero, we also prove that $\widetilde {\mathrm {VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the value group.

For a group G, a subgroup $U \leqslant G$ and a group A such that $\mathrm {Inn}(G) \leqslant A \leqslant \mathrm {Aut}(G)$, we say that U is an A-covering group of G if $G = \bigcup _{a\in A}U^a$. A theorem of Jordan (1872), implies that if G is a finite group, $A = \mathrm {Inn}(G)$ and U is an A-covering group of G, then $U = G$. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1990) conjectured that, more generally, there is an integer function f such that if G is a finite group and U is an A-covering subgroup of G, then $|G:U| \leqslant f(|A:\mathrm {Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger [‘Kronecker classes of fields and covering subgroups of finite groups’, J. Aust. Math. Soc. 57 (1994), 17–34], which asserts that there is a two-variable integer function g such that if G is a finite group and U is an A-covering subgroup of G, then $|G:U|\leqslant g(|A:\mathrm {Inn}(G)|,c)$, where c is the number of A-chief factors of G. Unfortunately, the proof of this theorem contains an error. In this paper, using a different argument, we give a correct proof of the theorem.

Pre-H-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-H-fields that are differential-Hensel–Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax–Kochen/Ershov theorem for such structures: the theory of a differential-Hensel–Liouville closed pre-H-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel–Liouville closed pre-H-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-H-field sort, from which we deduce that the ordered differential residue field is purely stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel–Liouville closed pre-H-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-H-fields with gap $0$, and is complete, distal, and locally o-minimal.

Let $p$ be a prime number and let $F$ be a field of characteristic different from $p$. We prove that there exist a field extension $L/F$ and $a,b,c,d$ in $L^{\times }$ such that $(a,b)=(b,c)=(c,d)=0$ in $\mathrm {Br}(L)[p]$ but the mod p Massey product $\langle a,b,c,d\rangle$ is not defined over $L$. Thus, the strong Massey vanishing conjecture at the prime $p$ fails for $L$, and the cochain differential graded ring $C^{* }(\Gamma _L,\mathbb Z/p\mathbb Z)$ of the absolute Galois group $\Gamma _L$ of $L$ is not formal. This answers a question of Positselski. As our main tool, we define a secondary obstruction that detects non-triviality of unramified torsors under tori, and which is of independent interest.

We show that if a field A is not pseudo-finite, then there is no prime model of the theory of pseudo-finite fields over A. Assuming GCH, we extend this result to $\kappa $-prime models, for $\kappa $ an uncountable cardinal or $\aleph _\varepsilon $.

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 $g(x)=x^3+ax^2+bx+c$ and $f(x)=g(x^3)$ be irreducible polynomials with rational coefficients, and let $ {\mathrm{Gal}}(f)$ be the Galois group of $f(x)$ over $\mathbb {Q}$. We show $ {\mathrm{Gal}}(f)$ is one of 11 possible transitive subgroups of $S_9$, defined up to conjugacy; we use $ {\mathrm{Disc}}(f)$, $ {\mathrm{Disc}}(g)$ and two additional low-degree resolvent polynomials to identify $ {\mathrm{Gal}}(f)$. We further show how our method can be used for determining one-parameter families for a given group. Also included is a related algorithm that, given a field $L/\mathbb {Q}$, determines when L can be defined by an irreducible polynomial of the form $g(x^3)$ and constructs such a polynomial when it exists.

In [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let M be an atomic and strongly $\omega $-homogeneous structure over a set of parameters A. Let B be a normal extension of A in M. We show that a short exact sequence of automorphism groups $1 \to \operatorname {\mathrm {Aut}}(M/B) \to \operatorname {\mathrm {Aut}}(M/A) \to \operatorname {\mathrm {Aut}}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in [4].

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 consider the relationship between the Mahler measure $M(f)$ of a polynomial f and its separation $\operatorname {sep}(f)$. Mahler [‘An inequality for the discriminant of a polynomial’, Michigan Math. J. 11 (1964), 257–262] proved that if $f(x) \in \mathbb {Z}[x]$ is separable of degree n, then $\operatorname {sep}(f) \gg _n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relationship and led to questions about the optimal exponent on $M(f)$. However, there has been relatively little study concerning upper bounds on $\operatorname {sep}(f)$ in terms of $M(f)$. We prove that if $f(x) \in \mathbb {C}[x]$ has degree n, then $\operatorname {sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$; for example, if it has only real roots.

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.