Estoy estudiando varios temas de matemáticas, y he topado con el muy buen libro "Abstract Algebra" de Celine Carstensen, Benjamin Fine, y Gerhard Rosenberg. Es un libro que toca muchos temas que me interesan. Leo hoy:
Abstract algebra or modern algebra can be best described as the theory of algebraic structures. Briefly, an algebraic structure is a set S together with one or more binary operations on it satisfying axioms governing the operations. There are many algebraic structures but the most commonly studied structures are groups, rings, fields and vector spaces. Also widely used are modules and algebras...
Ellos dividen en:
...Mathematics traditionally has been subdivided into three main areas – analysis, algebra and geometry. These areas overlap in many places so that it is often difficult to determine whether a topic is one in geometry say or in analysis. Algebra and algebraic methods permeate all these disciplines and most of mathematics has been algebraicized – that is uses the methods and language of algebra. Groups, rings and fields play a major role in the modern study of analysis, topology, geometry and even applied mathematics...
Los orígenes, por un lado la teoría de números:
Abstract algebra has its origins in two main areas and questions that arose in these areas – the theory of numbers and the theory of equations. The theory of numbers deals with the properties of the basic number systems – integers, rationals and reals while the theory of equations, as the name indicates, deals with solving equations, in particular polynomial equations. Both are subjects that date back to classical times. A whole section of Euclid"s elements is dedicated to number theory. The foundations for the modern study of number theory were laid by Fermat in the 1600s and then by Gauss in the 1800s. In an attempt to prove Fermat"s big theorem Gauss introduced the complex integers a C bi where a and b are integers and showed that this set has unique factorization. These ideas were extended by Dedekind and Kronecker who developed a wide ranging theory of algebraic number fields and algebraic integers. A large portion of the terminology used in abstract algebra, rings, ideals, factorization comes from the study of algebraic number fields. This has evolved into the modern discipline of algebraic number theory....
Por otro, la resolución de ecuaciones:
The second origin of modern abstract algebra was the problem of trying to determine a formula for finding the solutions in terms of radicals of a fifth degree polynomial. It was proved first by Ruffini in 1800 and then by Abel that it is imposible to find a formula in terms of radicals for such a solution. Galois in 1820 extended this and showed that such a formula is impossible for any degree five or greater. In proving this he laid the groundwork for much of the development of modern abstract algebra especially field theory and finite group theory. Earlier, in 1800, Gauss proved the fundamental theorem of algebra which says that any nonconstant complex polynomial equation must have a solution. One of the goals of this book is to present a comprehensive treatment of Galois theory and a proof of the results mentioned above...
Y aparece la geometría algebraica:
The locus of real points (x, y) which satisfy a polynomial equation f(x, y) = 0 is called an algebraic plane curve. Algebraic geometry deals with the study of algebraic plane curves and extensions to loci in a higher number of variables. Algebraic geometry is intricately tied to abstract algebra and especially commutative algebra.
Y cómo olvidar la resolución de sistemas de ecuaciones lineales:
Finally linear algebra, although a part of abstract algebra, arose in a somewhat different context. Historically it grew out of the study of solution sets of systems of linear equations and the study of the geometry of real n-dimensional spaces. It began to be developed formally in the early 1800s with work of Jordan and Gauss and then later in the century by Cayley, Hamilton and Sylvester.
En los próximos post, espero describri los contenidos de sus capítulos
Angel "Java" Lopez