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 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.