Splitting fields are associated with a polynomial, it is the smallest field in which polynomial splits. Since every polynomial splits in complex numbers, we just adjoin roots to base field to get the splitting field. If two fields are iso then we also have an iso for irreducible polynomial .
We also show existence of splitting fields via induction.
We now come to chain complex which is a fundamental tool used for solving problems. I cannot overstate its importance. In this lecture a large number of examples of chain complex are given.
We finally classify surfaces. Any compact connected surface is homeomorphic to a Sphere or connected sum of Torus or connected sum of projective planes. Whatever we cut we also paste it, this is simple rearrangement.
In this lecture we learn how to do simple constructions from CW complexes. There is a complete description of CW pair, Quotients, Wedge Sums, Product of Complexes and Smash Products with a lot of examples.
CW complexes are the basic building blocks of spaces. All, the basic concepts of Algebraic Topology can be understood and visualized via CW complexes. Homology, Cohomology, Cup Product are often easy to compute on these complexes, and a large number of qualifying/candidacy examination simply ask to compute these things on CW complexes.
One line slogan for constructing CW complex is
`Glue the boundary of n cell to n-1 cell
In the following two lectures I explain CW complexes and how they are built. You can go through the entire set of my videos on the YouTube channel.