With field coefficients homology and cohomology coincide With field coefficients the universal coefficients theorem takes the form:
$$H^n(X;F)=Hom_{F-modules}(H_n(X;F),F)$$.
Now in all computations I have seen with field coefficients we have
$$H^n(X;F)=H_n(X;F)$$. Is this true for any field or just for specific fields that we are usually working with, so my question can be reformulated as follows: for which fields $F$ do we have that $Hom_{F-modules}(H_n(X;F),F)=H_n(X;F)$? Thank you for your help!

Firstly, why are you writing $\hom_{F-modules}(H_n(X;F),F)$? UCT implies $$H^n(X;F)=\hom_{F}(H_n(X;F),F)$$.
Now, every $F$-module is a $F$-vector space and dual of a (finite dimensional) vector space is isomorphic to itself. So, $H^n(X;F)=\hom_F(H_n(X;F),F)=H_n(X;F)$, whenever $H_n(X;F)$ has finite rank.
For the infinite rank case see this SE post.
