Does there exist a simple proof of Universal Coefficient for the case of field coefficients?
For field $F$, we know that $$H^n(X;F)\cong Hom_F(H_n(X;F),F)$$
(Is homology with coefficients in a field isomorphic to cohomology?)
Is there a simplified proof for this special case, that is simpler than the whole proof involving Ext?
Thanks.
If you go through a standard proof, as in Hatcher, you will find a lot of shortcuts when doing things over a field. For instance $$0\to Z_n\to C_n\to B_{n-1}\to0$$ will split, as every exact sequence splits over a field. Dualising exact sequences gives exact sequences, so Hatcher's sequence (vii) will be exact, and no Ext terms are needed.
