The Theorem of the Primitive Element

2021 ◽  
Vol 128 (8) ◽  
pp. 753-754
Author(s):  
Steven H. Weintraub
Keyword(s):  
Primitive Element

Generalized synchronization via the differential primitive element

2014 ◽  
Vol 232 ◽  
pp. 848-857 ◽  
Author(s):  
Rafael Martínez-Guerra ◽  
Juan L. Mata-Machuca
Keyword(s):  
Primitive Element ◽  
Generalized Synchronization

scholarly journals Generalizations of the primitive element theorem

1991 ◽  
Vol 14 (3) ◽  
pp. 463-470
Author(s):  
Christos Nikolopoulos ◽  
Panagiotis Nikolopoulos
Keyword(s):  
Primitive Element ◽  
Infinite Field ◽  
Finitely Generated ◽  
Separable Algebras ◽  
Element Theorem

In this paper we generalize the primitive element theorem to the generation of separable algebras over fields and rings. We prove that any finitely generated separable algebra over an infinite field is generated by two elements and if the algebra is commutative it can be generated by one element. We then derive similar results for finitely generated separable algebras over semilocal rings.

scholarly journals One variable equations over semigroups

1970 ◽  
Vol 2 (2) ◽  
pp. 247-252 ◽  
Author(s):  
Frank Levin
Keyword(s):  
Free Product ◽  
Primitive Element ◽  
Single Variable ◽  
Several Variables ◽  
Separable Extensions

An analogue of the theorem on the existence of a primitive element for separable extensions of fields is presented for semigroups. This has two immediate consequences.(i) A semigroup is algebraically closed with respect to equations in several variables if and only if it is closed with respect to equations in a single variable.(ii) Any countable semigroup C is embedded in a two-generator semigroup, one of whose generators is in C.Further, a proof is given that any free product of a semigroup of order one with one of order two is SQ–universal, that is, its factor semigroups embed all countable semigroups. The proofs are adaptations of one used by Trevor Evans, Proc. Amer. Math. Soc. 3 (1952), 614–620. to show that a free product of two infinite cyclic semigroups is SQ–universal.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

2022 ◽  
Vol 29 (01) ◽  
pp. 151-166
Author(s):  
Himangshu Hazarika ◽  
Dhiren Kumar Basnet
Keyword(s):  
Positive Integer ◽  
Finite Fields ◽  
Primitive Element ◽  
Sufficient Condition ◽  
Extension Field ◽  
Cubic Form ◽  
Normal Element

For a prime [Formula: see text]and a positive integer[Formula: see text], let [Formula: see text] and [Formula: see text] be the extension field of [Formula: see text]. We derive a sufficient condition for the existence of a primitive element [Formula: see text] in[Formula: see text] such that [Formula: see text] is also a primitive element of [Formula: see text], a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is a primitive element of [Formula: see text], and a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is also a primitive normal element of [Formula: see text] over [Formula: see text].

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.

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