scholarly journals Field Extension by Galois Theory

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Md Taufiq Nasseef
Keyword(s):  
Galois Theory ◽  
Field Extension

scholarly journals GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP

2005 ◽  
Vol 16 (06) ◽  
pp. 567-593
Author(s):  
T. M. GENDRON ◽  
A. VERJOVSKY
Keyword(s):  
Galois Theory ◽  
Class Group ◽  
Number Fields ◽  
Field Extension ◽  
Abelian Extension ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Infinite Degree ◽  
Groups Of Automorphisms ◽  
Holomorphic Extensions

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 [Formula: see text] is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ𝖧•[K] of graded-holomorphic functions on [Formula: see text] 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 𝖢K is the idele class group, it is shown that there are embeddings of 𝖢K into 𝖦𝖺𝗅⊕(ℙ𝖧•[K ab ]/K) and 𝖦𝖺𝗅⊗(ℙ𝖧•[K ab ]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only.

ON CONJUGATE PRIME IDEALS OF TENSOR PRODUCTS OF k-ALGEBRAS

2010 ◽  
Vol 09 (01) ◽  
pp. 1-10 ◽  
Author(s):  
SAMIR BOUCHIBA
Keyword(s):  
Prime Ideal ◽  
Galois Theory ◽  
Tensor Products ◽  
Field Extension ◽  
Prime Ideals ◽  
Dimension Theory ◽  
Ideal Structure ◽  
Normal Field ◽  
Minimal Prime

The purpose of this paper is to explore new aspects of the prime ideal structure of tensor products of algebras over a field k. We prove that given a k-algebra A and a normal field extension K of k (in the sense of Galois theory), then for any prime ideals P1 and P2 of K ⊗k A lying over a fixed prime ideal p of A, there exists a k-automorphism σ of K such that (σ ⊗k id A)(P1) = P2. As an Application, we establish a result related to the dimension theory of tensor products stating that, for two arbitrary k-algebras A and B, the minimal prime ideals of p ⊗k B + Aσk q have the same height, for any prime ideals p and q of A and B, respectively.

scholarly journals Galoisian approach to complex oscillation theory of some Hill equations

2019 ◽  
Vol 124 (1) ◽  
pp. 102-131
Author(s):  
Yik-Man Chiang ◽  
Guo-Fu Yu
Keyword(s):  
Galois Theory ◽  
Oscillation Theory ◽  
Field Extension ◽  
Hill Equation ◽  
Fredholm Integral Equations ◽  
Heun Equation ◽  
Exponent Of Convergence ◽  
Closed Form Solutions ◽  
Liouvillian Solutions ◽  
Biconfluent Heun Equation

We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponent of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. These are solutions obtainable by suitable differential field extension constructions. In particular, we establish a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of a biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality, and satisfy Fredholm integral equations over suitable integration regions in $\mathbb{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.

scholarly journals A GALOIS THEORY FOR THE FIELD EXTENSION K((X))/K

2010 ◽  
Vol 52 (3) ◽  
pp. 447-451
Author(s):  
ANGEL POPESCU ◽  
ASIM NASEEM ◽  
NICOLAE POPESCU
Keyword(s):  
Power Series ◽  
Cyclic Group ◽  
Galois Theory ◽  
Finite Extension ◽  
Field Extension ◽  
Algebraic Extension ◽  
Finite Cyclic Group ◽  
Valued Field ◽  
Finite Subgroups ◽  
One To One

AbstractLet K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K((X)) be the valued field of Laurent power series and let G = Aut(F/K). We prove that if L is a subfield of F, K ≠ L, such that L/K is a sub-extension of F/K and F/L is a Galois algebraic extension (L/K is Galois coalgebraic in F/K), then L is closed in F, F/L is a finite extension and Gal(F/L) is a finite cyclic group of G. We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F/K. Some other auxiliary results which are useful by their own are given.

Field Extensions and Galois Theory

1984 ◽  
Author(s):  
Julio R. Bastida ◽  
Roger Lyndon
Keyword(s):  
Galois Theory ◽  
Field Extensions

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.

Galois theory of equations

1972 ◽  
pp. 243-266
Author(s):  
R. Kochendörffer
Keyword(s):  
Galois Theory

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

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