By Douglas R. Farenick (auth.)

The target of this publication is twofold: (i) to provide an exposition of the fundamental idea of finite-dimensional algebras at a levelthat isappropriate for senior undergraduate and first-year graduate scholars, and (ii) to supply the mathematical starting place had to organize the reader for the complex examine of someone of numerous fields of arithmetic. the topic below examine is under no circumstances new-indeed it truly is classical but a publication that provides a simple and urban remedy of this conception turns out justified for numerous purposes. First, algebras and linear trans formations in a single guise or one other are typical positive factors of varied elements of contemporary arithmetic. those contain well-entrenched fields resembling repre sentation conception, in addition to more recent ones comparable to quantum teams. moment, a research ofthe easy concept offinite-dimensional algebras is especially necessary in motivating and casting gentle upon extra subtle themes equivalent to module concept and operator algebras. certainly, the reader who acquires an exceptional knowing of the elemental idea of algebras is wellpositioned to ap preciate leads to operator algebras, illustration idea, and ring thought. In go back for his or her efforts, readers are rewarded by means of the consequences themselves, a number of of that are basic theorems of awesome elegance.

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7 Notes Hamilton, Grassmann, Dedekind, Frobenius, Peano, Hahn, Weyl, Banach, Noether, and von Neumann-these are some of the prominent mathematicians who have played a significant role in the developments that ultimately brought the axioms for a vector space into common usage. R. Hamilton published his now-celebrated paper on quaternions, a theory of linear equations was already well established, and the use of Cramer's rule and determinants (but not matrices) lay at its centre. Matrix theory actually came about after determinants, in large part through the efforts of E.

Let IF be any field. 1. There exists a field extension iF of IF such that (i) iF is an algebraic extension of IF, and (ii) iF is algebraically closed. Moreover, if IT{ is any algebraically closed algebraic extension of IF, then there is a field isomorphism T : IT{ ---t iF such that T(() = ( for all ( E IF. 2. If IT{ is a field extension of JR. , then IT{ and C are isomorphic fields. 6 Existence of Bases for Infinite-Dimensional Spaces This section is not a prerequisite for reading the subsequent chapters.

Banach's work demonstrated profoundly the value of studying abstract vector spaces in their own right , independent of the particular disciplines to which theorems about vector spaces might apply. As an analyst, Banach worked mostly with the field lR.. But a wide array of different fields are of interest in algebra. More generally, E. Artin and E. 8 EXERCISES 33 vector space concept, and what resulted was the notion of a module over a ring, which contained vector spaces as a special case. Thus, when in the early 1930s linear algebra at last arose as an independent subject area, it was devoted to the study of modules and their homomorphisms.