AlgTopReview: An informal introduction to abstract algebra

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• published: 10 Aug 2012
• views: 18396

This is a review lecture on some aspects of abstract algebra useful for algebraic topology. It provides some background on fields, rings and vector spaces for those of you who have not studied these objects before, and perhaps gives an overview for those of you who have. Our treatment is descriptive and informal; we sketch the main ideas and give some key examples, starting with the basic number systems of elementary arithmetic: the natural numbers, integers and rational numbers. Probably the easiest object is that of a field, with the rational numbers as the main example, although we also give an introduction to complex rational numbers, to finite fields (a non-standard approach) and to the rational numbers adjoined with an (algebraic!) square root of two. Examples of rings include the integers, polynomials and square matrices of a certain size. Examples of vectors spaces include row vectors, also polynomials up to a certain degree, and matrices of a fixed shape only with addition and scalar multiplication. The lecture ends with three prominent constructions familiar in almost all branches of algebra: the idea of subobjects, homomorphisms, and quotient objects. In our next lecture we will have a closer look at groups, both commutative and non-commutative, which are perhaps the most important algebraic objects in algebraic topology. Thanks to Nguyen Le for filming.