In this note we consider fractions of the form 1/m and their floating-point representation in various arithmetic bases. For instance, what is 1/7 in base 2005? And, what about 1/4? We give a simple algorithm to answer these questions. In addition, we discuss an extension of Midy's theorem whose proof relies on elementary modular arithmetic.
In this paper, we find all the generic polynomials for geometric $\ell$-cyclic function field extensions over the finite fields $\mathbb{F}_q$ where $q= p^n$, $p$ prime integer such that $q \equiv -1 \mod \ell$ and $(\ell , p)=1$.
We give a construction of the genus field for Kummer $\ell^n$-cyclic extensions of rational congruence function fields, where $\ell$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer $\ell$-cyclic extension. Finally, we study the extension $(K_1K_2)_{\frak{ge}}/(K_1)_{\frak{ge}}(K_2)_{\frak{ge}}$, for $K_1$, $K_2$ abelian extensions of $k$.
We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.
We produce generalizations of Iwasawa's `Riemann-Hurwitz' formula for number fields. These generalizations apply to cyclic extensions of number fields of degree p^n for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa λ-invariants which generalizes a result of Takashi Fukuda et. al. for totally real number fields.
The aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.
For all positive integers $\ell$, we prove non-trivial bounds for the $\ell$-torsion in the class group of $K$, which hold for almost all number fields $K$ in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-$p$-extensions of $F$, where $F$ is an arbitrary number field and $p$ is any prime for which $F$ and the $p$-th cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely…
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.
For a cyclic Kummer extension $K$ of a rational function field $k$ is considered, via class field theory, the extended Hilbert class field $K_H^+$ of $K$ and the corresponding extended genus field $K_g^+$ of $K$ over $k$, along the lines of the definitions of R. Clement for such extensions of prime degree. We obtain $K_g^+$ explicitly. Also, we use cohomology to determine the number of ambiguous classes and obtain a reciprocity law for $K/k$. Finally, we present a necessary and sufficient condition for a prime of $K$ to decompose fully in $K_g^+$.
For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky, and Kuwata: Let $n = 3,4,$ or $6$. If $K$ contains its $n^{th}$-roots of unity then, for any elliptic curve $E$ over $K$, there are infinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gains rank.
Upon quotienting by units, the elements of norm 1 in a number field $K$ form a countable subset of a torus of dimension $r_1 + r_2 - 1$ where $r_1$ and $r_2$ are the numbers of real and pairs of complex embeddings. When $K$ is Galois with cyclic Galois group we demonstrate that this countable set is equidistributed in a finite cover of this torus with respect to a natural partial ordering induced by Hilbert's Theorem 90.
Let $K$ be a global function field and fix a place $\infty$ of $K$. Let $L/K$ be a finite real abelian extension, i.e. a finite, abelian extension such that $\infty$ splits completely in $L$. Then we define a group of elliptic units $C_L$ in $\mathcal{O}_L^\times$ analogously to Sinnott's cyclotomic units and compute the index $[\mathcal{O}_L^\times:C_L]$. In the second part of this article, we additionally assume that $L$ is a cyclic extension of prime power degree. Then we can use the methods from Greither and Kučera to take certain roots of these elliptic units and prove a result on the ann…
In this paper, we investigate the existence of an elementary abelian closure in characteristic not $2$ for biquadratic extensions. We discover that it exists for any non-cyclic extension. We make use of it to obtain a classification for this class of extensions up to isomorphism via descent. This permits us to describe the geometry of this moduli space in group theoretic terms. We also provide two families of polynomials with two parameters that can describe any quartic extensions.
Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).
Let $q$ be a power of a prime number $p$, $k=\mathbb{F}_{q}(t)$ be the rational function field over finite field $\mathbb{F}_{q}$ and $K/k$ be a multi-cyclic extension of prime degree. In this paper we will give an exact formula for the degree of $K$ over $k$ by considering both Kummer and Artin-Schreier cases.
The local Oort conjecture states that, if G is cyclic and k is an algebraically closed field of characteristic p, then all G-extensions of k[[t]] should lift to characteristic zero. We prove a critical case of this conjecture. In particular, we show that the conjecture is always true when v_p(|G|) \leq 3, and is true for arbitrarily highly p-divisible cyclic groups G when a certain condition on the higher ramification filtration is satisfied.
Let K/Q be a cyclic extension. In this paper, we give several congruences connecting the prime divisors of the degree g= [K:Q] with the prime divisors of the class number h of K/Q. As an exemple, the theorem: Let K/Q be a cyclic extension with [K:Q]=g. Suppose that g is not divisible by 2 . Let g_j, j=1,...m, be the prime divisors of g. Let h_i, i=1,...r, be the prime divisors of the class number h of K/Q. If for one prime factor h_i of h, the h_i-component G(h_i) of the class group G of K/Q is cyclic then: else h_i divides g, else h_i = 1 (mod g_j) for at least one prime divisor g_j of g. The…
In cyclic, degree 8 extensions of algebraic number fields $N/K$, ambiguous ideals in N are canonical $\mathbb{Z}[C_8]$-modules. Their $\mathbb{Z}[C_8]$-structure is determined here. It is described in terms of indecomposable modules and determined by ramification invariants. Although infinitely many indecomposable $\mathbb{Z}[C_8]$-modules are available (classification by Yakovlev), only 23 appear.
For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.
We study the ramification of fierce cyclic Galois extensions of a local field $K$ of characteristic zero with a one-dimensional residue field of characteristic $p>0$. Using Kato's theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs $(δ_i,ω_i)$, where $δ_i$ is a positive rational number and $ω_i$ a differential form on the residue field of $K$. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of $K$.
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 $K[C_{p^n}]$.
Let $p\neq2$ be a prime. We show a technique based on local class field theory and on the expansions of certain resultants which allows to recover very easily Lbekkouri's characterization of Eisenstein polynomials generating cyclic wild extensions of degree $p^2$ over $\Q_p$, and to extend it to the case of the base field $K$ being an unramified extension of $\Q_p$. Furthermore, when a polynomial satisfies only some of the stated conditions, we show that the first unsatisfied condition gives information about the Galois group of the normal closure. This permits to give a complete classificatio…
Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete charac…
We compute the ramification filtration on wildly ramified $p^2$-cyclic extensions of local fields of characteristic $p$. The ramification filtration on the compositum of two $p$-cyclic and $p^2$-cyclic extensions are also computed. As an application, some partial results towards Abhyankar's Inertia conjecture has been proved.
The aim of this paper is to study certain family of elliptic curves $\{\mathscr{X}_H\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathscr{X}/F$ associated to a cyclic cubic extension $K/F$. We show that each $\mathscr{X}_H$ admits a 3-isogeny $ϕ$ over $F$ and the dual Selmer group $S^{(\hatϕ)}(\hat{\mathscr{X}_H}/F)$ is bounded by a kind of unit/class groups attached to $K/F$. This is proven via certain rational function on the elliptic curve $\mathscr{X}_H$ with nice property. We also prove that the Shafarevich-Tate group $\text{\cyr X} (\hat{\…
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D varies over cube-free positive integers. We parametrise these points using well known parametrisation of integral points (x,y,z) of the curve X^2+3Y^2=4Z^3 with GCD(y,z)=1. As an accidental byproduct of our result we show that there are infinitely many primes congruent to 1 or 8 modulo 9, can be expressed as sum of two rational cubes.
Suppose G is a semi-direct product of the form Z/p^n \rtimes Z/m where p is prime and m is relatively prime to p. Suppose K is a local field of characteristic p > 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration whose dimension depends only on the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p^3.
Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we obtain bounds for the F_ rank of the l-Sylow subgroup of K using genus theory. We obtain some results valid for general l. Following that, we obtain more complete results for l=5 and F =Q(ζ_5). The rank of the 5-class group of K is expressed in terms of power residue symbols. We compare our results with tables obtained using SAGE (the latter is under GRH). We …
Let $p$ be prime, and $n,m \in \mathbb{N}$. When $K/F$ is a cyclic extension of degree $p^n$, we determine the $\mathbb{Z}/p^m\mathbb{Z}[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times p^m}$. With at most one exception, each indecomposable summand is cyclic and free over some quotient group of $\text{Gal}(K/F)$. For fixed values of $m$ and $n$, there are only finitely many possible isomorphism classes for the non-free indecomposable summand. These Galois modules act as parameterizing spaces for solutions to certain inverse Galois problems, and therefore this module computation provid…
- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the …
We show that for any square-free natural number $n$ and any global field $K$ with $(\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\in K^*\times K^*$ such that $x$ is not a norm of $K(\sqrt[n]{y})/K$ form a diophantine set over $K$. We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that for any $n\in \mathbb{N}$ and any global field $K$ with $(\text{char}(K), n)=1$, $K^*\setminus K^{*n}$ is diophantine over $K$. For a number field $K$, this is a result of Colliot-Thélène and Van Geel, proved using results on the…
In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on …
We study fluctuations in the number of points of $\ell$-cyclic covers of the projective line over the finite field $\mathbb{F}_q$ when $q \equiv 1 \mod \ell$ is fixed and the genus tends to infinity. The distribution is given as a sum of $q+1$ i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick, while statistics were obtained for certain components of the moduli space of $\ell$-cyclic covers by Bucur, David, Feigon and Lalín. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of $…
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 $0$.
This work presents author's explicit methods of constructing abelian extensions of complete discrete valuation fields. His approach to explicit equations of a cyclic extension of degree p^n which contains a given cyclic extension of degree p is explained.
We give a construction of unramified cyclic octic extensions of certain complex quadratic number fields. The binary quadratic form used in this construction also shows up in the theory of 2-descents on Pell conics and elliptic curves, as well as in the explicit description of cyclic quartic extensions.
Let p be a prime and suppose that K/F is a cyclic extension of degree p^n with group G. Let J be the F_pG-module K^*/K^{*p} of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p^i+1 for some 0<=i<n. We show that for all primes p and all 0<=i<n, there exists a field extension K/F with a summand of dimension p^i+1.
For every finite $p$-group $G_p$ that is non-cyclic and non-quaternion and every positive integer $\ell\neq p$ that is greater than $2$, we prove the first non-trivial bound on $\ell$-torsion in class group of every $G_p$-extension. More generally, for every nilpotent group $G$ where every Sylow-$p$ subgroup $G_p\subset G$ is non-cyclic and non-quaternion, we prove a non-trivial bound on $\ell$-torsion in class group of every $G$-extension for every integer $\ell>1$.
We show the cardinality of a particular nonabelian cohomology associated to cyclic division algebras is equal to a certain partition number. This computation helps us interpret the number of conjugacy classes of maximal tori defined over an abelian extension.
By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular.
$\DeclareMathOperator{\Aut}{Aut}$Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $Γ$ are groups of order $p^{2} q$ with non-is…
Let K/F be a cyclic field extension of odd prime degree. We consider Galois embedding problems involving Galois groups with common quotient Gal(K/F) such that corresponding normal subgroups are indecomposable Fp[Gal(K/F)]-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.
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…
