The norm function of an algebraic field extension

AbstractLet X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

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Filter Rings Under Flat Base Change

Algebra Colloquium ◽

10.1142/s1005386708000448 ◽

2008 ◽

Vol 15 (03) ◽

pp. 463-470

Author(s):

Parviz Sahandi ◽

Siamak Yassemi

Keyword(s):

Field Extension ◽

Base Change ◽

Prime Ideals ◽

Algebraic Field ◽

Flat Base ◽

Minimal Prime ◽

Local Homomorphism

Let φ: (R, 𝔪) → (S, 𝔫) be a flat local homomorphism of rings. In this paper, we prove: (1) If dim S/𝔪S > 0, then S is a filter ring if and only if R and k(𝔭) ⊗R𝔭 S𝔮 are Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭= 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. (2) If dim S/𝔪S = 0, then S is a filter ring if and only if R is a filter ring and k(𝔭) ⊗R𝔭 S𝔮 is Cohen–Macaulay for all 𝔮 ∈ Spec (S) \ {𝔫} and 𝔭 = 𝔮 ∩ R, and S/𝔭S is catenary and equidimensional for all minimal prime ideals 𝔭 of R. As an application, it is shown that for a k-algebra R and an algebraic field extension K of k, if K ⊗k R is locally equidimensional, then R is a locally filter ring if and only if K ⊗k R is a locally filter ring.

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The norm function of an algebraic field extension. II

Pacific Journal of Mathematics ◽

10.2140/pjm.1955.5.519 ◽

1955 ◽

Vol 5 (4) ◽

pp. 519-528 ◽

Cited By ~ 2

Author(s):

Harley Flanders

Keyword(s):

Field Extension ◽

Algebraic Field

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Separability in an Algebra with Semi-Linear Homomorphism

Canadian Journal of Mathematics ◽

10.4153/cjm-1972-062-3 ◽

1972 ◽

Vol 24 (4) ◽

pp. 668-671

Author(s):

David J. Winter

Keyword(s):

Associative Algebra ◽

Identity Element ◽

Field Extension ◽

Simple Theory ◽

Algebraic Field ◽

Finite Dimensional ◽

Power Operation

The purpose of this paper is to outline a simple theory of separability for a non-associative algebra A with semi-linear homomorphism σ. Taking A to be a finite dimensional abelian Lie p-algebra L and σ to be the pth power operation in L, this separability is the separability of [2]. Taking A to be an algebraic field extension K over k and σ to be the Frobenius (pth power) homomorphism in K, this separability is the usual separability of K over k. The theory also applies to any unital non-associative algebra A over a field k and any unital homomorphism σ from A to A such that σ(ke) ⊂ ke, e being the identity element of A.

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On the Integral Degree of Integral Ring Extensions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000275 ◽

2018 ◽

Vol 62 (1) ◽

pp. 25-46

Author(s):

José M. Giral ◽

Liam O'Carroll ◽

Francesc Planas-Vilanova ◽

Bernat Plans

Keyword(s):

Field Extension ◽

Integral Closure ◽

Ring Extension ◽

Algebraic Field ◽

Integral Domains ◽

Upper Semicontinuous ◽

Number Of Generators ◽

Integrally Closed ◽

Finite Integral ◽

Integral Degree

AbstractLet A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

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

Jurnal Sains Komunikasi dan Pengembangan Masyarakat [JSKPM] ◽

10.29244/jskpm.4.4.545-560 ◽

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.

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Normal Bases on Galois Ring Extensions

Symmetry ◽

10.3390/sym10120702 ◽

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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