- Objectives of the course (generalities)
This is a first course on what has been usually terme Abstract Algebra, whose objective is to introduce some of the fundamental abstract objects (concepts, tools, ...) of algebra. Those ideas originated in a huge variety of situations in mathematics, physics and other disciplines, as it is apparent in its widespread presence in Mathematics and its applications. Among those sources of algebraic concepts we can stress the study of natural numbers (and its generalizations), which leads to Arithmetics, and more abstractly, to Number Theory. Our course starts with some arithmetic motivation which gives rise, by abstraction, to the notion if ring, a concept whose conceptual power manifests not only in Number Theory, but also in almost every area of Mathematics and Physics. So, as a preparation for that, we first recall some basic results of the Arithmetics of Integers and apply them to some results on what is probably the central theme of (algebraic) Number Theory, namely, the study of diophantine equations. We then address the "arithmetic" of general (though mainly commutative) rings, so examining the behavior of divisibility, and the means or relating different rings through the notion of homomorphism (and its particular instances, isomorhisms, epimorphisms and monomorphisms). In a final part, we deal with the notion of module, a generalization of that of vector space, which exemplifies a general fact in Mathematics, the strength of linearization. Most often we will restrict ourselves to commutative rings and modules over commutative rings.
The course is divided into the following main sections:
0. Integers and Arithmetics.
1. Rings
3. Modules
(Some) References:
The bibliography of the subject is huge, so we will only mention some very classical textbooks.
N. Bourbaki, Elements of Mathematics. Algebra, (2 vol.) Springer.
P. M. Cohn, Groups, Rings and Fields, Springer
N. Jacobson, Basic algebra I and II, Dover Publications, inc.
S. Lang, Algebra, Springer.
B. L. van der Waerden, Modern algebra, 2 vols. Springer.
- Teacher: Montaner Frutos, Fernando