scholarly journals Galois extensions in which every element with regular trace is a normal basis element

1958 ◽  
Vol 9 (2) ◽  
pp. 222-222
Author(s):  
Carl C. Faith
Keyword(s):  
Normal Basis ◽  
Galois Extensions ◽  
Basis Element

Cleft extensions and Galois extensions for Hom-associative algebras

2016 ◽  
Vol 27 (03) ◽  
pp. 1650025 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Associative Algebra ◽  
Galois Extension ◽  
Classical Case ◽  
Normal Basis ◽  
Associative Algebras ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

scholarly journals Cleft extensions for a Hopf algebrakq[X,X−1, Y]

1998 ◽  
Vol 40 (2) ◽  
pp. 147-160 ◽  
Author(s):  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Primitive Element ◽  
Normal Basis ◽  
Crossed Products ◽  
Galois Extensions ◽  
Infinite Dimensional ◽  
Cleft Extensions ◽  
Cocommutative Hopf Algebra

The concept of cleft extensions, or equivalently of crossed products, for a Hopf algebra is a generalization of Galois extensions with normal basis and of crossed products for a group. The study of these subjects was founded independently by Blattner-Cohen-Montgomery [1] and by Doi-Takeuchi [4]. In this paper, we determine the isomorphic classes of cleft extensions for a infinite dimensional non-commutative, non-cocommutative Hopf algebra kq[X, X–l, Y], which is generated by a group-like element X and a (1,X)-primitive element Y. We also consider the quotient algebras of the cleft extensions.

scholarly journals H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS

2017 ◽  
Vol 21 (21) ◽  
pp. 23-23
Author(s):  
J. N. Alonso Alvarez ◽  
J. M. Fernandez Vilaboa ◽  
R. Gonzalez Rodriıguez
Keyword(s):  
Hopf Algebras ◽  
Normal Basis ◽  
Galois Extensions ◽  
Weak Hopf Algebras

scholarly journals Weak quasi-entwining structures

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6229-6252
Author(s):  
Álvarez Alonso ◽  
Vilaboa Fernádez ◽  
González Rodríguez
Keyword(s):  
Galois Extension ◽  
Hopf Algebras ◽  
Normal Basis ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

In this paper we introduce the notion of weak quasi-entwining structure as a generalization of quasi-entwining structures and weak entwining structures. Also, we formulate the notions of weak cleft extension, weak Galois extension, and weak Galois extension with normal basis associated to a weak quasientwining structure. Moreover, we prove that, under some suitable conditions, there exists an equivalence between weak Galois extensions with normal basis and weak cleft extensions. As particular instances, we recover some results previously proved for Hopf quasigroups, weak Hopf quasigroups and weak Hopf algebras.

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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