Narrow Results

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over ℚ, a hyperbolized adele class group is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ𝖧_•[K] of graded-holomorphic functions on possesses two operations ⊕ and ⊗ giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that 𝖦𝖺𝗅(ℙ𝖧_•[K]/K) = 1 and 𝖦𝖺𝗅(ℙ𝖧_•[L]/ℙ𝖧_•[K]) ≅ 𝖦𝖺𝗅(L/K) if L/K is Galois. If K^ab denotes the maximal abelian extension of K and 𝖢_K is the idele class group, it is shown that there are embeddings of 𝖢_K into 𝖦𝖺𝗅_⊕(ℙ𝖧_•[K^ab]/K) and 𝖦𝖺𝗅_⊗(ℙ𝖧_•[K^ab]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only.

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M. Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida’s formula on $\lambda$-invariants in a $p$-extension of ${\mathbb{Z}}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa’s second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched ${\mathbb{Z}}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.

The aim of this paper is to present some connections of infinite Cogalois Theory with Clifford extensions and Hopf algebras.

A formula for computing the discriminant of any number field K ⊂ ℚ(ζ_2^r), with r ≥ 3, is derived. The formula consists of two expressions, depending on whether K is cyclotomic or not. However, both expressions depend solely on m, the degree of K over ℚ, and they are derived from the Conductor-Discriminant Formula for Abelian extensions of ℚ.

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory.

Let p be a prime number and let ℚ/ℤ′ be the elements in ℚ/ℤ of order prime to p. Let , where c is a prime power of p. We use characters and valuation theory to prove that Δ is a parameter space for the cyclic tame extensions of the formal Laurent series field k_p((t)) of degree prime to p. Furthermore, we construct the cyclic tame extension corresponding to a given triple in Δ. The structure of finite cyclic tame extensions of the p-adic number fields was thoroughly investigated by A. A. Albert in 1935. Here we get the same result as consequence of our main theorem.

For a proper subfield K of we show the existence of an algebraic number α such that no power αⁿ, n ≥ 1, lies in K. As an application it is shown that these numbers, multiplied by convenient Gaussian numbers, can be written in the form P(T)^Q(T) for some transcendental numbers T where P and Q are arbitrarily prescribed nonconstant rational functions over .

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu )$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2p^2$, we determine exactly the possible sets of Hopf Galois structure types.

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#Â$. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.

A well known result of Schur states that if n is a positive integer and a₀, a₁,…,a_n are arbitrary integers with a₀a_n coprime to n!, then the polynomial is irreducible over the field ℚ of rational numbers. In case each a_i = 1, it is known that the Galois group of f_n(x) over ℚ contains A_n, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials f_n(x) which leads to the construction of trinomials of degree n for each n with Galois group S_n, the symmetric group on n letters.

A set of real nth roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.

For f(x) ∈ ℤ[x] and a ∈ ℤ, we let fⁿ(x) be the nth iterate of f(x), P(f, a) = {p prime: p|fⁿ(a) for some n}, and D(P(f, a)) denote the natural density of P(f, a) within the set of primes. A conjecture of Jones [5] indicates that D(P(f, a)) = 0 for most quadratic f. In this paper, we find an exceptional family of (f, a) such that D(P(f, a)) > 0 by considering f_t(x) = (x + t)² - 2 - t and a_t = f_t(0) for t ∈ ℤ. We prove that if t is not of the form ±M² ± 2 or ±2M² ± 2, then D(P(f_t, a_t)) = ⅓. We also determine D(P(f_t, a_t)) in some cases when the density is not equal to ⅓. Our results suggest a connection between the arithmetic dynamics of the conjugates of x² and the conjugates of x² - 2.

In this paper, we show that if p ≡ 1 (mod 4) is prime, then F_p admits a representation of the form u² + pv² for some integers u and v, where F_n is the nth Fibonacci number.

In this article, we investigate systems of fundamental units of quartic fields which are constructed by adjoining a root of a certain parametric quartic polynomial to ℚ.

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions $1+\mathrm{\text{log}}(1-x)$ and $\mathrm{\text{cos}}x$. We first show that the Galois groups of the $n$th Taylor polynomials of $1+\mathrm{\text{log}}(1-x)$ are as large as possible, namely, $S_n$ (full symmetric group) or $A_n$ (alternating group), depending on the residue of the integer number $n$ modulo $4$. We then compute the Galois groups of the $n$th Taylor polynomials of $\mathrm{\text{cos}}(x)$ and show that these Galois groups essentially coincide with the Coexter groups of type $B_n$ (or an index 2 subgroup of the corresponding Coexter group).

We present a family of algorithms for computing the Galois group of a polynomial defined over a p-adic field. Apart from the “naive” algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of ${\mathbb{Q}}_2$ of degrees 18, 20 and 22, tables of which are available online.

Let $X$ be a curve defined over $\mathbb{Q}$ and let $t\in \mathbb{Q}(X)$ be a non-constant rational function on $X$ of degree $v\ge 2$. For every rational number $a/b$ pick a point $P_{a/b}\in X(\overline{\mathbb{Q}})$ such that $t(P_{a/b})=a/b$. In this paper, we obtain lower bounds on the number of distinct fields among $\mathbb{Q}(P_{a/b})$ with $1\le a,b\le N$ under some assumptions on $t$. We show that if $t$ has a pole of order at least 2 or if there is a rational number $\alpha$ such that $t-\alpha$ has a zero of order at least 2, then the set $\{\mathbb{Q}(P_{a/b})|1\le a,b\le N\}$ contains $\gg \frac{N^2}{{(logN)}^2}$ elements. We also obtain partial results when $t$ does not have a pole of order at least two.

For a prime p, positive integers $r,n$, and a polynomial f with coefficients in ${𝔽}_{p^r}$, let $W_{p,r,n}(f)=f^n\left({𝔽}_{p^r}\right)\backslash f^{n+1}\left({𝔽}_{p^r}\right)$. As n varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of f on ${𝔽}_{p^r}$. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of ${𝔽}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages.

We continue with our study of vague groups. We also examine the notion of a vague field. For a pair (X, μ), where μ is a vague binary operation on X, we define in the language of fuzzy equalities, the notion of an identity and of an inverse of an element in X. We show that these definitions are equivalent to the original definitions of an identity and inverse of an element in a vague group. We further show how a vague group can be constructed using an ascending chain of normal subgroups. This construction of vague groups is transitive of first and second order [3]. We show that a preimage of a vague group under a homomorphism f is a ker f-vague group. Construction of a vague field using a descending chain of subfields is introduced.