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Galois extensions and $$O^{*}$$-fields
Positivity, 2023zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Kenneth Evans, Jingjing Ma
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2015
A small portion of linear algebra and combinatorics are used in the development of Hamming codes, the first generation error control codes. The design of error control codes such as BCH codes and Reed Solomon codes relies on the structures of Galois fields and polynomials over Galois fields.
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A small portion of linear algebra and combinatorics are used in the development of Hamming codes, the first generation error control codes. The design of error control codes such as BCH codes and Reed Solomon codes relies on the structures of Galois fields and polynomials over Galois fields.
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1984
If F is an algebraic number field of degree n over Q and p is a prime, then F is p-realizable if there is a tor-sionfree abelian group A of rank n such that qA = A for all prines q i p and F is isomorphic to the quasi-endomorphism algebra of A. The question “for which F and p is F p-realizable?” was the subject of the paper by Pierce and Vinsonhaler ...
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If F is an algebraic number field of degree n over Q and p is a prime, then F is p-realizable if there is a tor-sionfree abelian group A of rank n such that qA = A for all prines q i p and F is isomorphic to the quasi-endomorphism algebra of A. The question “for which F and p is F p-realizable?” was the subject of the paper by Pierce and Vinsonhaler ...
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Reconfigurable Galois Field multiplier
2014 International Symposium on Biometrics and Security Technologies (ISBAST), 2014Galois Field has received a lot of attention because of their important and particular applications in cryptography, channel coding, etc. This paper presents the Reconfigurable Galois Field multiplier used to calculate the Galois field multiplication of different lengths which consists of AND gates and special cells.
Rong-Jian Chen +2 more
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HOPF–GALOIS STRUCTURES ON FIELD EXTENSIONS WITH SIMPLE GALOIS GROUPS
Bulletin of the London Mathematical Society, 2003Let \(L/K\) be a Galois extension of fields with Galois group \(G\). Then the group ring \(KG\) endows \(L/K\) with a Hopf Galois structure. There may be other \(K\)-Hopf algebras \(H\) endowing \(L/K\) with a Hopf Galois structure; all of them are forms of group rings over \(L\).
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Field Theory of Galois' Fields
1995The motivation for the present work comes from various sources which, however, need not be elaborated on here. I will be exploring a class of quantum field theories defined over finite sets of integers. Essentially these are the familiar Z„ lattice theories, but carried to their logical extremes.
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1995
Consider an algebraic system \(\langle SF, +, \cdot \rangle\) consisting of a finite set of elements, \(SF\) with two binary operations, addition and multiplication, \(|SF|=q\), \(q\) an arbitrary integer \(\geq 2\). A spurious Galois field, \(SGF(q)\) satisfies, in addition, the following axioms: A.1.
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Consider an algebraic system \(\langle SF, +, \cdot \rangle\) consisting of a finite set of elements, \(SF\) with two binary operations, addition and multiplication, \(|SF|=q\), \(q\) an arbitrary integer \(\geq 2\). A spurious Galois field, \(SGF(q)\) satisfies, in addition, the following axioms: A.1.
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