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ALGEBRAIC VARIETIES OVER FIELDS WITH DIFFERENTIATION
Mathematics of the USSR-Sbornik, 1969It is known that there do not exist algebraic homomorphisms of the multiplicative group of a field into the additive group . However, if the field has a nontrivial differentiation , then the logarithmic derivative gives a homomorphism , .Ju. I. Manin observed that for abelian varieties over a field with a nontrivial differentiation it is possible to ...
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2010
18.1. Let us first recall the notion of an algebra over a field that we introduced in §11.1. By an algebra over a field F, or simply by an F -algebra, we understand an associative ring A which is also a vector space over F such that \((ax)(by)=abxy\) for \(\,a,\,b\in F\) and \(\,x,\,y\in A.\) If A has an identity element, we denote it by \(1_A,\) or ...
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18.1. Let us first recall the notion of an algebra over a field that we introduced in §11.1. By an algebra over a field F, or simply by an F -algebra, we understand an associative ring A which is also a vector space over F such that \((ax)(by)=abxy\) for \(\,a,\,b\in F\) and \(\,x,\,y\in A.\) If A has an identity element, we denote it by \(1_A,\) or ...
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1973
This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A)
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This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A)
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Factoring Polynomials over Algebraic Number Fields
SIAM Journal on Computing, 1985The author describes an algorithm for factoring polynomials over arbitrary number fields. This algorithm works as follows. Given a polynomial f, defined over a number field K. We take the norm N f of f to \({\mathbb{Q}}[X]\), and factor N f over \({\mathbb{Q}}\). If N f is square free we derive a factorization of f.
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Decomposition of algebras over finite fields and number fields
Computational Complexity, 1991zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Integral Models of Algebraic Tori Over Fields of Algebraic Numbers
Journal of Mathematical Sciences, 2016The paper is concerned with integral models of algebraic tori over algebraic number fields together with the comperison of properties of the models. It depends on the scheme which represent the (Néron, Voskresenskií) model under consideration. The author offers a setting for the Néron model of an algebraic torus, namely the standard or the canonical ...
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Groups of Algebras Over an Algebraic Number Field
American Journal of Mathematics, 1943MacLane, S., Schilling, O. F. G.
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QUANDLE G-ALGEBRA OVER A FIELD
JP Journal of Algebra, Number Theory and Applications, 2020Alghamdi, Ahmad M., Alfadhli, Amani M.
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1967
In this Chapter, k will be an A-field; we use all the notations introduced for such fields in earlier Chapters, such as k A , k v , r v , etc. We shall be principally concerned with a simple algebra A over k; as stipulated in Chapter IX, it is always understood that A is central, i. e. that its center is k, and that it has a finite dimension over k; by
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In this Chapter, k will be an A-field; we use all the notations introduced for such fields in earlier Chapters, such as k A , k v , r v , etc. We shall be principally concerned with a simple algebra A over k; as stipulated in Chapter IX, it is always understood that A is central, i. e. that its center is k, and that it has a finite dimension over k; by
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Sparse Algebraic Equations over Finite Fields
SIAM Journal on Computing, 2009A system of algebraic equations over a finite field is called sparse if each equation depends on a low number of variables. Efficiently finding solutions to the system is an underlying hard problem in cryptanalysis of modern ciphers. In this paper the deterministic Agreeing-Gluing algorithm introduced earlier by Raddum and Semaev for solving such ...
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