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.

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Classification of Surfaces

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.