SOME EXAMPLES IN COGALOIS THEORY WITH APPLICATIONS TO ELEMENTARY FIELD ARITHMETIC

2002 ◽  
Vol 01 (01) ◽  
pp. 1-29 ◽  
Author(s):  
TOMA ALBU
Keyword(s):  
Rational Numbers ◽  
Field Extension ◽  
Type Formula ◽  
Finite Sum ◽  
Elementary Field

The aim of this paper is to provide some examples in Cogalois Theory showing that the property of a field extension to be radical (resp. Kneser, or Cogalois) is not transitive and is not inherited by subextensions. Our examples refer especially to extensions of type [Formula: see text]. We also effectively calculate the Cogalois groups of these extensions. A series of applications to elementary arithmetic of fields, like: • for what n, d ∈ ℕ* is [Formula: see text] a sum of radicals of positive rational numbers • when is [Formula: see text] a finite sum of monomials of form [Formula: see text], where r, j1,…, jr ∈ ℕ*, c ∈ ℚ*, and [Formula: see text] are also presented.

Division probability for quaternion algebras over fields

2020 ◽  
pp. 2150098
Author(s):  
Leena Jindal ◽  
Anjana Khurana
Keyword(s):  
Rational Numbers ◽  
Equivalence Classes ◽  
Quaternion Algebras ◽  
Type Formula ◽  
Elementary Type ◽  
Witt Equivalence

Let [Formula: see text] be a field of [Formula: see text] with finitely many square classes. In this paper, we define a new rational valued invariant of [Formula: see text], and call it the division probability of [Formula: see text]. We compute it for all fields of elementary type. Further, we show that [Formula: see text], where [Formula: see text] is the number of Witt-equivalence classes of fields with [Formula: see text], and [Formula: see text] is the count of rational numbers that appear as division probabilities for fields [Formula: see text] of elementary type with [Formula: see text]. In the paper, we also determine [Formula: see text] for all [Formula: see text] and show that rational numbers of type [Formula: see text] always occur as division probability for a suitable field [Formula: see text].

scholarly journals Automatic realization of Hopf Galois structures

2020 ◽  
pp. 2250030
Author(s):  
Teresa Crespo
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Commutator Subgroup ◽  
Field Extension ◽  
Nonabelian Group ◽  
Type Formula ◽  
Galois Structure ◽  
Number Formula

We consider Hopf Galois structures on a separable field extension [Formula: see text] of degree [Formula: see text], for [Formula: see text] an odd prime number, [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] has at most one abelian type of Hopf Galois structures. For a nonabelian group [Formula: see text] of order [Formula: see text], with commutator subgroup of order [Formula: see text], we prove that if [Formula: see text] has a Hopf Galois structure of type [Formula: see text], then it has a Hopf Galois structure of type [Formula: see text], where [Formula: see text] is an abelian group of order [Formula: see text] and having the same number of elements of order [Formula: see text] as [Formula: see text], for [Formula: see text].

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

Productly linearly independent sequences

2015 ◽  
Vol 52 (3) ◽  
pp. 350-370
Author(s):  
Jaroslav Hančl ◽  
Katarína Korčeková ◽  
Lukáš Novotný
Keyword(s):  
Rational Numbers ◽  
Infinite Products ◽  
Linearly Independent ◽  
New Concepts ◽  
Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

Faktor-Faktor yang Berhubungan dengan Kinerja Penyuluh Pertanian di Kabupaten Ciamis Jawa Barat

2020 ◽  
Vol 4 (4) ◽  
pp. 545
Author(s):  
Nur Puti Kurniawati ◽  
Dwi Sadono ◽  
Endang Sri Wahyuni
Keyword(s):  
Achievement Motivation ◽  
Census Data ◽  
Work Motivation ◽  
Agricultural Extension ◽  
Field Extension ◽  
Need For Achievement ◽  
Extension Agent ◽  
Spearman Correlation Test ◽  
And Performance ◽  
The Relationship

Agricultural extension agent was the main spearhead in carrying out counseling. A good agricultural extension agent can be reflected in their performance. The purpose of this study were: (1) describe the characteristics of agricultural extension agent, (2) describe the level of competence, level of work motivation, and describe level of performance of agricultural extension agent, (3) analyze the relationship between characteristics of agricultural extention agent and the level of performance of agricultural extension agent, (4) analyze the relationship between the level of competency of agricultural extension agent and the level of performance of agricultural extension agent, (5) analyze the relationship between the level of motivation of agricultural extension agent and the level of performance of agricultural extension agent. Responden in this study were 48 field extension agent who are civil servant in Ciamis Regency West Java and selected by census. Data were analyzed using Rank Spearman correlation test. The results showed that agricultural extension agent in Ciamis Regency were dominated by extension agent who were old, undergraduate educated, had little training, and had a long working period. Agricultural extension agent in Ciamis Regency generally have sufficient competency which tends to be high and generally dominated by the need for achievement motivation. The results also show that there is a relationship between managerial competence and performance, social competence with performance, technical competence with performance, level of competency with performance, and the need for achievement with performance.Keywords: Agricultural Extension Agent,Competence, Motivation, Performance.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

scholarly journals Verlinde formulae on complex surfaces: K-theoretic invariants

2021 ◽  
Vol 9 ◽  
Author(s):  
L. Göttsche ◽  
M. Kool ◽  
R. A. Williams
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Complex Surfaces ◽  
Type Formula ◽  
Verlinde Formula ◽  
Donaldson Invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

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