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ARRIETA, E. A. Absolute Valued Algebraic Algebras. 2012. 120 f. Dissertação (Mestrado) Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2012. Our goal here is to study the absolute valued algebraic real algebras. In order to reach our intention, we regard an absolute valued real algebra and on which one we impose: First, such one is nite-dimensional algebra; second; such one is algebraic algebra; third, such one is with unity; and in the end such one is algebraic algebra. In the latter case, our aim, it needs of certain classic results of functional analysis and others one of Banach algebras; then, we reach that such one real algebra is isotope to one of the classical absolute valued real algebras R, C, H or D. Where H is the Quaternions real algebra and D is the Octonions real algebra. The absolute valued algebras are not necessarily associative. Abraham Adrian Albert was the rst mathematician considering absolute valued algebras in a context not necessarily associative. In 1947, he proved that any nite-dimensional absolute valued real algebra with unit element is isomorphic to either real eld R, the complex eld C, the Quaternions algebra H or the Octonions algebra D. Two years later, he demonstrated that R, C, H and D are the unique absolute valued algebraic real algebras with unit element. Recently, in 1997, Kaidi El-Amin, Maria Isabel Ramírez and Ángel Rodríguez Palacios proved that any absolute valued algebraic real algebra is nite-dimensional.

Álgebras algébricas absolutamente valuadas