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On Matrix Algebras Over an Algebraically Closed Field
The Annals of Mathematics, 1942Recently a number of writers have discussed interesting developments in the theory of not completely reducible matrix sets and non-semisimple algebras.' Here we have made use of some of these concepts and methods to study matrix algebras over an algebraically closed field.
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Factoring Polynomials Over Algebraic Number Fields
ACM Transactions on Mathematical Software, 1976A method for factoring polynomials whose coefficients are in an algebraic number field is presented. This method is a natural extension of the usual Henselian technique for factoring polynomials with integral coefficients. In addition to working in any number field, our algorithm has the advantage of factoring nonmonic polynomials without inordinately ...
Peter J. Weinberger +1 more
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Some Remarks on Algebras Over an Algebraically Closed Field
The Annals of Mathematics, 1943The theory of rings with radicals is an interesting and far reaching problem of modern algebra.' In this paper we have examined some aspects of algebras which may have radicals and whose coefficient fields are algebraically closed. Some of the methods employed clearly could be used for less restricted algebras, but a full extension of the results ...
Nesbitt, C., Scott, W. M.
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1995
Up to now we have not considered the possibility of multiplying two vectors to obtain another vector, though we have noted that this is possible in certain cases. For example, we can multiply elements of the vector space ℳ n×n (F) over a field F. A vector space V over a field F is an algebra over F if and only if there exists a bilinear transformation (
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Up to now we have not considered the possibility of multiplying two vectors to obtain another vector, though we have noted that this is possible in certain cases. For example, we can multiply elements of the vector space ℳ n×n (F) over a field F. A vector space V over a field F is an algebra over F if and only if there exists a bilinear transformation (
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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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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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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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Simple Algebras Over Rational Function Fields
Canadian Journal of Mathematics, 1979The well-known Hasse-Brauer-Noether theorem states that a simple algebra with center a number field k splits over k (i.e., is a full matrix algebra) if and only if it splits over the completion of k at every rank one valuation of k. It is natural to ask whether this principle can be extended to a broader class of fields.
Nyman, T., Whaples, G.
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