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The Theorem of the Primitive Element
The American Mathematical Monthly, 2021The theorem of the primitive element is one of the basic results of Galois theory. We present a proof, different from the standard one, that is undoubtedly not new, but which in our opinion deserve...
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k-Primitivity and images of primitive elements
Journal of Algebra and Its Applications, 2016Let [Formula: see text] be a free Lie algebra of finite rank [Formula: see text] [Formula: see text]. We give another proof of the following criterion which is proven by Mikhalev and Zolotykh, using the idea of [Formula: see text]-primitivity: An endomorphism of [Formula: see text] preserving primitivity of elements is an automorphism.
Ögüşlü N.Ş., Ekici N.
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Primitive and Almost Primitive Elements of Schreier Varieties
Journal of Mathematical Sciences, 2019According to the classical monograph by \textit{P. M. Cohn} [Universal Algebra. New York etc.: Harper and Row (1965; Zbl 0141.01002)], a variety of universal algebras is a class of all algebras that satisfy all identities from a given set of identities.
Artamonov, V. A. +3 more
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Tame Almost Primitive Elements
Results in Mathematics, 2000Let \(G\) be a finitely generated group, \(V\) a set of laws, and \(\mathcal V\) the variety defined by \(V\). An element \(\omega\in G\) is \(\mathcal V\)-generic in \(G\) if \(\omega\in V(G)\) but \(\omega\notin V(K)\) for any proper subgroup ...
Konieczny, Jochen +2 more
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Primitive elements with prescribed trace
Applicable Algebra in Engineering, Communication and Computing, 2014zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Xiwang Cao, Peipei Wang
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2007
Let n be a positive integer. A nonzero element γ of the finite field F of order q = 2n is said to be "strongly primitive" if every element (aγ+b)/(cγ+d), with a, b, c, d in {0, 1} and ad-bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to θθ′ ċ q where θ is the product of (1-1/p ...
Daniel Goldstein, Alfred W. Hales
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Let n be a positive integer. A nonzero element γ of the finite field F of order q = 2n is said to be "strongly primitive" if every element (aγ+b)/(cγ+d), with a, b, c, d in {0, 1} and ad-bc not zero, is primitive in the usual sense. We show that the number N of such strongly primitive elements is asymptotic to θθ′ ċ q where θ is the product of (1-1/p ...
Daniel Goldstein, Alfred W. Hales
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Primitive Elements in Symmetric Algebras
Canadian Journal of Mathematics, 1974Let-R be a commutative ring with 1, and let be the symmetric algebra of an R-module M. We regard the isomorphisms S 0(M) ≅ R and S
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Subgroups of free groups and primitive elements
Journal of Group Theory, 2010Let \(F\) be a free group on the basis \(X=\langle x_1,x_2,\dots,x_n\rangle\) and \(H\) be a finitely generated subgroup of \(F\). A question is [see \url{http://www.grouptheory.info/}, Question F39b]: Is it possible to decide if \(H\) contains a primitive element of \(F\)?
Clifford, A., Goldstein, R. Z.
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