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In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras.The theorem states that an (Artinian) [a] semisimple ring R is isomorphic to a product of finitely many n i-by-n i matrix rings over division rings D i, for some integers n i, both of which are uniquely determined up to permutation of the index i.
The Jordan–Chevalley decomposition of an element in algebraic group as a product of semisimple and unipotent elements; The Bruhat decomposition = of a semisimple algebraic group into double cosets of a Borel subgroup can be regarded as a generalization of the principle of Gauss–Jordan elimination, which generically writes a matrix as the product of an upper triangular matrix with a lower ...
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
The vector space of complex-valued class functions of a group has a natural -invariant inner product structure, described in the article Schur orthogonality relations.Maschke's theorem was originally proved for the case of representations over by constructing as the orthogonal complement of under this inner product.
Algorithms are known for decomposing univariate polynomials in polynomial time. Polynomials which are decomposable in this way are composite polynomials ; those which are not are indecomposable polynomials or sometimes prime polynomials [ 1 ] (not to be confused with irreducible polynomials , which cannot be factored into products of polynomials ).
President-elect Donald Trump speaks to members of the media following a meeting at the US Capitol on Jan. 8, 2025 in Washington, DC. Credit - Valerie Plesch—Bloomberg via Getty Images
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