Splitting Field

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 \varphi:F_1\simeq F_2 then we also have an iso \frac{F_1[x]}{p(x)}\simeq\frac{F_2[x]}{\varphi p(x)} for irreducible polynomial p(x).

We also show existence of splitting fields via induction.

Introducing CW Complexes

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 X^n to n-1 cell X^{n-1}

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.