# 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.

In this lecture I define the operation of Connected Sum of two surfaces $\Large{S_1}$ and $\Large{S_2}$. Take a disc out of $\Large{S_1}$ and a disc out of $\Large{S_2}$, and glue the boundary. This is an important operation and will be used in classification of two surfaces.