The Hopf–Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group G correspond bijectively with certain subgroups of $ \mathrm{Perm}(G) $. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf–Galois structures, which we term ρ-conjugation. We study properties of this construction, with particular emphasis on the Hopf–Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct ρ-conjugates of a given Hopf–Galois structure is determined by the corresponding skew left brace, and that if $ H, H^{\prime} $ are Hopf algebras giving ρ-conjugate Hopf–Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ \mathfrak{B} $ of L is free over its associated order in H if and only if it is free over its associated order in Hʹ. We exhibit a variety of examples arising from interactions with existing constructions in the literature.

Let F be a subfield of the complex numbers and $f(x)=x^6+ax^5+bx^4+cx^3+bx^2+ax+1 \in F[x]$ an irreducible polynomial. We give an elementary characterisation of the Galois group of $f(x)$ as a transitive subgroup of $S_6$. The method involves determining whether three expressions involving a, b and c are perfect squares in F and whether a related quartic polynomial has a linear factor. As an application, we produce one-parameter families of reciprocal sextic polynomials with a specified Galois group.

For a Galois extension $K/F$ with $\text {char}(K)\neq 2$ and $\mathrm {Gal}(K/F) \simeq \mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}$ , we determine the $\mathbb {F}_{2}[\mathrm {Gal}(K/F)]$ -module structure of $K^{\times }/K^{\times 2}$ . Although there are an infinite number of (pairwise nonisomorphic) indecomposable $\mathbb {F}_{2}[\mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/2\mathbb {Z}]$ -modules, our decomposition includes at most nine indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representation theory allows for an infinite number of indecomposable types.

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.

Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation $\theta \colon G\to \mathrm {GL}_1(\mathbb {Z}_p)$ such that every open subgroup H of G, together with the restriction $\theta \vert _H$ , satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally, we ask whether 1-smooth pro-p groups satisfy a “Tits’ alternative.”

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 $m_i$ .

It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

A system of quadratic forms is associated to every generalised quadratic form over a division algebra with involution of the first kind in characteristic two. It is shown that this system determines the isotropy behaviour and the isometry class of generalised quadratic forms. An application of this construction to the Witt index of generalised quadratic forms is also given.

Let $f(x)=x^{6}+ax^{4}+bx^{2}+c$ be an irreducible sextic polynomial with coefficients from a field $F$ of characteristic $\neq 2$, and let $g(x)=x^{3}+ax^{2}+bx+c$. We show how to identify the conjugacy class in $S_{6}$ of the Galois group of $f$ over $F$ using only the discriminants of $f$ and $g$ and the reducibility of a related sextic polynomial. We demonstrate that our method is useful for producing one-parameter families of even sextic polynomials with a specified Galois group.

In this paper, we will prove that any $\mathbb{A}^{3}$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of Kambayashi on the triviality of separable $\mathbb{A}^{2}$-forms over a field $k$ extends to $\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing $\mathbb{Q}$.

We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F, or more generally, of a bounded PAC field F. This paper answers some of the questions of [1], and in particular that any finite group which is geometrically represented in a pseudo-finite field must be abelian.

Let $P\in \mathbb{F}_{2}[z]$ be such that $P(0)=1$ and degree $(P)\geq 1$. Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory73 (1998), 292–317] proved that there exists a unique subset ${\mathcal{A}}={\mathcal{A}}(P)$ of $\mathbb{N}$ such that $\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$, where $p({\mathcal{A}},n)$ is the number of partitions of $n$ with parts in ${\mathcal{A}}$. Let $m$ be an odd positive integer and let ${\it\chi}({\mathcal{A}},.)$ be the characteristic function of the set ${\mathcal{A}}$. Finding the elements of the set ${\mathcal{A}}$ of the form $2^{k}m$, $k\geq 0$, is closely related to the $2$-adic integer $S({\mathcal{A}},m)={\it\chi}({\mathcal{A}},m)+2{\it\chi}({\mathcal{A}},2m)+4{\it\chi}({\mathcal{A}},4m)+\cdots =\sum _{k=0}^{\infty }2^{k}{\it\chi}({\mathcal{A}},2^{k}m)$, which has been shown to be an algebraic number. Let $G_{m}$ be the minimal polynomial of $S({\mathcal{A}},m)$. In precedent works there were treated the case $P$ irreducible of odd prime order $p$. In this setting, taking $p=1+ef$, where $f$ is the order of $2$ modulo $p$, explicit determinations of the coefficients of $G_{m}$ have been made for $e=2$ and 3. In this paper, we treat the case $e=4$ and use the cyclotomic numbers to make explicit $G_{m}$.

We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$. When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$

In this article we develop a test to determine whether a sextic polynomial that is irreducible over $\mathbb{Q}$ has Galois group isomorphic to the alternating group ${{A}_{4}}$. This test does not involve the computation of resolvents, and we use this test to construct several infinite families of such polynomials.

Let E/k be a function field over an infinite field of constants. Assume that E/k(x) is a separable extension of degree greater than one such that there exists a place of degree one of k(x) ramified in E. Let K/k be a function field. We prove that there exist infinitely many nonisomorphic separable extensions L/K such that [L:K]=[E:k(x)] and AutkL=AutKL≅Autk(x)E.

We consider the classical universal covering $\exp: {{\mathbb C}}\To {{\mathbb C}}^*$ of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion-free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of ${{\mathbb C}}$. This can also be seen as a model-theoretic statement on the categoricity of a corresponding $L_{\omega_1,\omega}$-sentence. The proof is a combination of arithmetic and model-theoretic methods.

We give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

If $K$ is an algebraic function field of one variable over an algebraically closed field $k$ and $F$ is a finite extension of $K$, then any element $a$ of $K$ can be written as a norm of some $b$ in $F$ by Tsen's theorem. All zeros and poles of $a$ lead to zeros and poles of $b$, but in general additional zeros and poles occur. The paper shows how this number of additional zeros and poles of $b$ can be restricted in terms of the genus of $K$, respectively $F$. If $k$ is the field of all complex numbers, then we use Abel's theorem concerning the existence of meromorphic functions on a compact Riemann surface. From this, the general case of characteristic 0 can be derived by means of principles from model theory, since the theory of algebraically closed fields is model-complete. Some of these results also carry over to the case of characteristic $p>0$ using standard arguments from valuation theory.

Let $L/K$ be a finite Galois extension of fields whose Galois group is a nonabelian simple group. It is shown that $L/K$ admits exactly two Hopf–Galois structures.

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.