arXiv subjects
Publications and source records attributed to Daniel Gil-Muñoz.
Let $p$ be an odd prime number. For a degree $p$ extension of $p$-adic fields $L/K$, we give a complete characterization of the condition for the ring of integers $\mathcal{O}_L$ to be free as a module over its associated order in the unique Hopf-Galois structure on $L/K$.
For a cubic number field $L$, we consider the $\mathbb{Z}$-order in $L$ of the form $\mathbb{Z}[α]$, where $α$ is a root of a polynomial of the form $x^3-ax+b$ and $a,b\in\mathbb{Z}$ are integers such that $v_p(a)\leq 2$ or $v_p(b)\leq 3$ for all prime numbers $p$. We characterize the freeness of $\mathbb{Z}[α]$ as a module over its associated order in the unique Hopf-Galois structure $H$ on $L$ in terms of the solvability of at least one between two generalized Pell equations in terms of $a$ and $b$. We determine when the equality $\mathcal{O}_L=\mathbb{Z}[α]$ is satisfied in terms of congruence conditions for $a$ and $b$. For such cases, we specialize our result so as to obtain criteria for the freeness of $\mathcal{O}_L$ as a module over its associated order in $H$.
We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an integer $n$ satisfying $p\nmid n$, $p\nmid|\mathrm{Aut}(E)|$ for every group $E$ of order $n$ and such that each group of order $np$ has a unique $p$-Sylow subgroup. Under these conditions, we prove that any skew left brace of size $np$ is either a twofold semidirect product of the trivial brace of size $p$ and a skew left brace of size $n$ or a companion skew left brace of that one. We develop an algorithm to obtain all skew left braces of size $np$ from the skew left braces of size $n$ and provide a formula to count them. We use this result to describe all skew left braces of size $12p$ for $p\geq 7$, which proves a conjecture of V.G. Bardakov, M.V. Neshchadim and M.K. Yadav.
We consider Shanks' simplest cubic fields $K$ for which the index $[\mathcal{O}_K:\mathbb{Z}[ρ]]$ of a root $ρ$ of the defining parametric polynomial is $3$. For them, we study the additive indecomposables of $K$ and provide a complete list of them. Moreover, we use the knowledge of the indecomposables to prove some interesting consequences on the arithmetic of $K$. Mainly, we obtain good bounds on the ranks of universal quadratic forms over $K$ and prove that the Pythagoras number of $\mathcal{O}_K$ is $6$.
We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an $H$-Galois extension $L/K$ is $H$-Kummer if $L$ can be generated by adjoining to $K$ a finite set $S$ of eigenvectors for the action of the Hopf algebra $H$ on $L$. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of $H$-Kummer extensions $L/K$ as radical extensions that are linearly disjoint with the $n$-th cyclotomic extension of $K$. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the $n$-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions $L_1/K$, $L_2/K$ such that $L_1\cap M_2=L_2\cap M_1=K$, where $M_i$ is a field such that $L_iM_i=\widetilde{L}_i$, the normal closure of $L_i/K$. When $L/K$ is an extension of number or $p$-adic fields, we shall derive criteria on the freeness of the ring of integers $\mathcal{O}_L$ over its associated order in an almost classically Galois structure on $L/K$.
We describe all left braces of size 8p for p an odd prime different from 3 or 7 and validate the number given by Bardakov, Neschadim and Yadav. We give a characterization for isomorphism classes of a semidirect product of left braces and then the description is done by first describing left braces of size 8, as conjugacy classes of regular subgroups of the corresponding holomorph, and then checking how many non isomorphic left braces of size 8p are obtained from each one of them.
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mathfrak{A}_{L/K}$-module.
Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and such that every group of order $np$ has a normal subgroup of order $p$. (This holds in particular for $p>n$.) We prove that left braces of size $np$ may be obtained as a semidirect product of the unique left brace of size $p$ and a left brace of size $n$. We give a method to determine all braces of size $np$ from the braces of size $n$ and certain classes of morphisms from the multiplicative group of these braces of size $n$ to $\mathrm{Z}_p^*$. From it we derive a formula giving the number of Hopf Galois structures of abelian type $\mathrm{Z}_p \times E$ on a Galois extension of degree $np$ in terms of the number of Hopf Galois structures of abelian type $E$ on a Galois extension of degree $n$. For a prime number $p\geq 7$, we apply the obtained results to describe all left braces of size $12p$ and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree $12p$.
The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.
Given a quartic Galois extension $L/\mathbb{Q}$ of number fields and a Hopf-Galois structure $H$ on $L/\mathbb{Q}$, we study the freeness of the ring of integers $\mathcal{O}_L$ as module over the associated order $\mathfrak{A}_H$ in $H$. For the classical Galois structure $H_c$, we know by Leopoldt's theorem that $\mathcal{O}_L$ is $\mathfrak{A}_{H_c}$-free. If $L/\mathbb{Q}$ is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in $\mathbb{Z}$ of certain generalized Pell equations. We shall translate some results on Pell equations into results on the $\mathfrak{A}_H$-freeness of $\mathcal{O}_L$.
Let $p$ be an odd prime. For field extensions $L/\mathbb{Q}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and the module structure of the ring of integers $\mathcal{O}_L$. We solve the case in which $L/\mathbb{Q}_p$ is not totally ramified and present a practical method which provides a complete answer for the cases $p=3$ and $p=5$. We see that within this family of dihedral extensions, the ring of integers is always free over the associated orders in the different Hopf Galois structures.