Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and $\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a profinite Lie $\mathbb F_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. The canonical $\mathcal P$-topology on $\mathcal K$ is used to define a dense Lie subalgebra $\mathcal L^{\mathcal P}$ in $\mathcal L$. The algebra $\mathcal L^{\mathcal P}$ can be provided with a system of $\mathcal P$-topological generators and its $\mathcal P$-open subalgebras correspond to all $N$-dimensional extensions of $\mathcal K$ in $\mathcal K_{<p}$. These results are applied to higher local fields $K$ of characteristic 0 containing primitive $p$-th root of unity. If $Γ=\mathop{Gal}(K_{alg}/K)$ we introduce similarly the quotient $Γ_{<p}=G(L)$, a dense $\mathbb F_p$-Lie algebra $L^{\mathcal P}\subset L$, and describe the structure of $L^{\mathcal P}$ in terms of generators and relations. The general result is illustrated by explicit presentation of $Γ_{<p}$ modulo third commutators.
The fundamental theorem of arithmetic factorizes any integer into a product of prime numbers. The Jordan-Holder theorem dissolves many groups by their normal series which can be refined into composition series. The main topic of this thesis is the dissociation of field extensions. We dissociate algebraic extensions by their intermediate fields to build a tower with the greatest possible number of Galois steps. We call "galtowerable" the extensions admitting a field tower all the steps of which are Galois extensions,i.e. a "Galois tower". Two Galois towers of the same galtowerable extension (fi…
Let $L/K$ be a finite Galois extension of local or global fields in characteristic $0$ or $p$ with nonabelian Galois group $G$, and let ${\mathfrak B}$ be a $G$-stable fractional ideal of $L$. We show that ${\mathfrak B}$ is free over its associated order in $K[G]$ if and only if it is free over its associated order in the Hopf algebra giving the canonical nonclassical Hopf-Galois structure on the extension.
Recently, much work has been done to investigate Galois module structure of local field extensions, particularly through the use of Galois scaffolds. Given a totally ramified $p$-extension of local fields $L/K$, a Galois Scaffold gives us a $K$-basis for $K[G]$ whose effect on the valuation of elements of $L$ is easy to determine. In 2013, N.P. Byott and G.G. Elder gave sufficient conditions for the existence of Galois scaffolds for cyclic extensions of degree $p^2$ in characteristic $p$. We take their work and adapt it to cyclic extensions of degree $p^2$ in characteristic $0$.
A Galois scaffold, in a Galois extension of local fields with perfect residue fields, is an adaptation of the normal basis to the valuation of the extension field, and thus can be applied to answer questions of Galois module structure. Here we give a sufficient condition for a Galois scaffold to exist in fully ramified Galois extensions of degree $p^2$ of characteristic $p$ local fields. This condition becomes necessary when we restrict to $p=3$. For extensions $L/K$ of degree $p^2$ that satisfy this condition, we determine the Galois module structure of the ring of integers by finding necessa…
We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois extension can be prescribed has been investigated by Dèbes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois extensions arising as specializations of a family of regular Galois extensions.
Let $K$ be a local field and let $L/K$ be a totally ramified Galois extension of degree $p^n$. Being semistable and possessing a Galois scaffold are two conditions which facilitate the computation of the additive Galois module structure of $L/K$. In this note we show that $L/K$ is semistable if and only if $L/K$ has a Galois scaffold. We also give sufficient conditions in terms of Galois scaffolds for the extension $L/K$ to be stable.
Given a field $k$ and a finite group $H$, {\it{an $H$-parametric extension over $k$}} is a finite Galois extension of $k(T)$ of Galois group containing $H$ which is regular over $k$ and has all the Galois extensions of $k$ of group $H$ among its specializations. We are mainly interested in producing non $H$-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann-Black problem and the existence of one parameter generic polynomials. We develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples…
This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, con…
Given an arbitrary field $F$, we describe all Galois extensions $L/F$ whose Galois groups are isomorphic to the group of upper triangular unipotent 4-by-4 matrices with entries in the field of two elements.
Let $L/k$ an Galois extension of number fields with Galois group isomorphic to a dihedral group of order $2n$. In this note, we give a general description of the Hasse norm principle for $L/k$ and the weak approximation for the norm one torus $R^1_{L/k}(\mathbb{G}_m)$ associated to $L/k$.
We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has Galois group permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions"). In particular, for alternating groups and (an infinite family of) projective linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$ gain rank over infinitely many $G$-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion which allows to deduce that "many" elliptic curves gain rank over infin…
We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of $ p $-adic fields or number fields which is $ H $-Galois for a commutative Hopf algebra $ H $. Firstly, we show that if $ L/K $ is a tame Galois extension of $ p $-adic fields then each fractional ideal of $ L $ is free over its associated order in $ H $. We also show that this conclusion remains valid if $ L/K $ is merely almost classically Galois. Finally, we show that if $ L/K $ is an abelian extension of number fields then every ambiguous fractional ideal of $ L…
Nigel Boston and Barry Mazur have shown how to determine the natural subspaces of certain S_3-extensions of the rationals, which they term "generic". We extend some of their results to another class of extensions, called "degenerate".
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{…
For fields F of characteristic not p containing a primitive $p$th root of unity, we determine the Galois module structure of the group of $p$th-power classes of K for all cyclic extensions K/F of degree p.
We introduce and study a class of field extensions that we call pre-Galois; viz. extensions that become Galois after some linearly disjoint Galois base change. Among them are geometrically Galois extensions of k(T), with k a field: extensions that become Galois and remain of the same degree after passing to the algebraic closure of k. We develop a pre-Galois theory that includes a Galois correspondence, and investigate the corresponding variants of the inverse Galois problem. We provide answers in situations where the classical analogs are not known. In particular, for every finite simple grou…
Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with Galois group $G$ among its specializations. We are mainly interested in producing non-$G$-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann-Black problem. Building on a strategy developed in previous papers, we show that there exists at least one non-$G$-parametric extension over $F$ for a given non-trivial…
Ramification groups of Galois extensions over local fields of positive characteristic with Galois group isomorphic to the group of unitriangular matrices
We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This property is known for all solvable groups due to Shafarevich's solution of the inverse Galois problem for those groups. It is however completely open for nonsolvable groups. In this paper, we provide general criteria to attack such questions via specialization of function field extensions, and in particular give the first infinite families of Galois realiza…
Let $k\subseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $\mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $\underline{G}\subset\mathscr{G}$ with the property that $Spec_k(K)$ is a $k$-torsor for $\underline{G}$. $\underline{G}$ is a constant $k$-group if and only if $G$ is abelian, in which case $G=\underline{G}$.
We generalize a result of F.\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field $K$, then for any degree $k\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field…
Let $L/K$ be a primitive purely inseparable extension of fields of characteristic $p$, $\left[ L:K\right] >p.$ It is well known that $L/K$ is Hopf Galois for some Hopf algebra $H$, and it is suspected that $L/K$ is Hopf Galois for numerous choices of $H$. We construct a family of $K$-Hopf algebras $H$ for which $L$ is an $H$-Galois object. For some choices of $K$ we will exhibit an infinite number of such $H.$ We provide some explicit examples of the dual, Hopf Galois, structure on $L/K.$
This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\Q$ and over $\F_q(t)$ ramified only at one finite prime. Over $\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\F_q(t)$ ramified only at one prime.
Let L_1 and L_2 be finite separable extensions of a global field K, and let E_i be the Galois closure of L_i over K for i=1,2. We establish a local-global principle for the product of norms from L_1 and L_2 (so-called multinorm principle) provided that the extensions E_1 and E_2 are linearly disjoint over K.
By Langlands and Deligne we know that the local constants are extendible functions. Therefore, to give an explicit formula of the local constant of an induced representation of a local Galois group of a non-Archimedean local field $F$ of characteristic zero, we have to compute the lambda function $λ_{K/F}$ for a finite extension $K/F$. In this paper, when a finite extension $K/F$ is Galois, we give a formula for $λ_{K/F}$.
The conjecture due to Bertrand and Rodriguez Villegas asserts that the 1-norm of the nonzero element in an exterior power of the units of a number field has a certain lower bound. We prove this conjecture for the exterior square case when the number field is a real multi-quadratic Galois extension of any degree of the rationals.
Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers $O_L$ to be free of rank 1 over its associated order $A_0$, and to stricter conditions which imply that $A_0$ is a Hopf order in the group ring $K[C_{p^n}]$.
We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local function fields of characteristic $p$, called {\em one-dimensional}, that, in a particular sense, are as simple as cyclic degree $p$ extensions, and prove the statement in the title above.
In the mid-1960s Borevic and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F.
Let $ L/K $ be a finite separable extension of local or global fields in any characteristic, let $ H_{1}, H_{2} $ be two Hopf algebras giving Hopf-Galois structures on the extension, and suppose that the actions of $ H_{1}, H_{2} $ on $ L $ commute. We show that a fractional ideal $ {\mathfrak B} $ of $ L $ is free over its associated order in $ H_{1} $ if and only if it is free over its associated order in $ H_{2} $. We also study which properties these associated orders share.
We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8.
Let $ L/K $ be a finite separable extension of fields whose Galois closure $ E/K $ has group $ G $. Greither and Pareigis have used Galois descent to show that a Hopf algebra giving a Hopf-Galois structure on $ L/K $ has the form $ E[N]^{G} $ for some group $ N $ such that $ |N|=[L:K] $. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as $ K $-algebras. By applying our results, we complete a detailed analysis…
We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.
Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. We obtain an upper bound for the number of degree $|F|$ algebraic extensions $K/k$ with Galois group $G$ with the norm of the discriminant $\mathcal{N}_{k/\mathbb{Q}}(d_{K/k})$ bounded above by $X$. We extend this method for any group $G$ that has an abelian normal subgroup. If $G$ has an abelian normal subgroup, then we obtain upper bounds for the number of degree $|G|$ extensions $N/k$ with Galois group $G$ with bounded norm of the discriminant. Malle made a conjecture about w…
For every prime $p\geq 5$ for which a certain condition on the class group $\text{Cl}(\mathbb{Q}(μ_p))$ is satisfied, we construct a $p$-adic analytic Galois extension of the infinite cyclotomic extension $\mathbb{Q}(μ_{p^{\infty}})$ with some special ramification properties. In greater detail, this extension is unramified at primes above $p$ and tamely ramified above finitely many rational primes and is isomorphic to a finite index subgroup of $\text{SL}_2(\mathbb{Z}_p)$ which contains the principal congruence subgroup. For the primes $107,139,271$ and $379$ such extensions were first constru…
Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any finiteness hypotheses. We show that these groups are finite and obtain, in particular, formulas for the corresponding quotient of group orders.
We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$ consisting of unipotent four by four matrices over ${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condit…
In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to …
For given rational prime number $p$ consider the tower of finite extensions of fields $K_0/\mathbb{Q}_p,$ $K/K_0, L/K, M/L$, where $K/K_0$ is unramified and $M/L$ is a Galois extension with Galois group $G$. Suppose one dimensional Honda formal group over the ring $\mathcal{O}_K$, relative to the extension $K/K_0$ and uniformizer $π\in K_0$ is given. The operation $x\underset{F}+y=F(x,y)$ sets a new structure of abelian group on the maximal ideal $\mathfrak{p}_M$ of the ring $\mathcal{O}_M$ which we will denote by $F(\mathfrak{p}_M)$. In this paper the structure of $F(\mathfrak{p}_M)$ as $\mat…
In this paper we construct families of rotated $D_n$-lattices which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels via subfields of cyclotomic fields. These constructions exhibit full diversity and good minimum product distance which are important parameters related to the signal transmission error probability. It is also shown that for some Galois extensions $\matK|\matQ$ it is impossible to construct rotated $D_n$-lattices via fractional ideals of $\mathcal O_{\mattK}$.
The Polya group of a number field K is the subgroup of the class group of K generated by the classes of the products of the maximal ideals with same norm. A Polya field is a number field whose Polya group is trivial. Our purpose is to start with known assertions about Polya fields to find results concerning Polya groups. This first paper inspired by Zantema's results focuses on the Polya group of the compositum of two galoisian extensions of Q.
Given a finite group G and a number field k, a well-known conjecture asserts that the set R_t(k,G) of Steinitz classes of tame G-Galois extensions of k is a subgroup of the ideal class group of k. In this paper we investigate an explicit candidate for R_t(k,G), when G is of odd order. More precisely, we define a subgroup W(k,G) of the class group of k and we prove that R_t(k,G) is contained in W(k,G). We show that equality holds for all groups of odd order for which a description of R_t(k,G) is known so far. Furthermore, by refining techniques introduced in arXiv:0910.5080v1, we use the Shafar…
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 $\mat…
Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.
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…
We disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups $G$ which do not have a realization $F/\Qq(T)$ that induces all Galois extensions $L/\Qq(U)$ of group $G$ by specializing $T$ to $f(U) \in \Qq(U)$. For these groups, we produce two extensions $L/\Qq(U)$ that cannot be simultaneously induced, thus even disproving a weaker Lifting Property. Our examples of such groups $G$ include symmetric groups $S_n$, $n\geq 7$, infinitely many $PSL_2(\Ff_p)$, the Monster. Two variants of the question with $\Qq(U)$ replaced by $\Cc(U)$ and $\Qq$ are answered similarly, t…
We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is an unramified extension of $ p $-adic fields which is $ H $-Galois for some Hopf algebra $ H $ then $ \OL $ is free over its associated order $ \AH $ in $ H $. If $ H $ is commutative, we show that this conclusion remains valid in ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. By combining these results we prove a generalisation of Noether's th…
We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.
