Why are elliptic curves constructed using prime fields and not composite fields? I come across this:


  Numbers mod composite number does not form a field rather it forms a ring


and


  every number has a multiplicative inverse under integer mod prime


Maybe these are the reasons why prime field is preferred? By the first above fact it means that construction of a composite field is not possible. But I have seen some research articles about multiplication algorithm using composite fields…

Why are elliptic curves constructed using prime fields and not composite fields?
For a prime $p$ and an integer $n\geq1$, the ring $\mathbb{Z}/p^n\mathbb{Z}$ is a field if and only if $n=1$. There are fields with $p^n$ elements, usually denoted $\mathbb{F}_{p^n}$ or $\operatorname{GF}(p^n)$, but they are constructed differently. For $n\ge2$, they are commonly called extension fields (as opposed to prime fields for $n=1$), as they can be viewed as extensions of $\mathbb{F}_p$.

There are examples of elliptic curves over extension fields. For example, every cryptographically secure curve in characteristic two is defined over a field of the form $\mathbb{F}_{2^m}$, i.e. a field with $2^m$ elements. There is also the Four$\mathbb{Q}$ curve, which is defined over $\mathbb{F}_{p^2}$ where $p=2^{127}-1$. 

Doing things over extension fields can get a bit more tricky. There are attacks which can exploit the fact that $n\ge2$, which lead to faster algorithms for the discrete logarithm problem, on which elliptic-curve crypto is built. This gets complex pretty quickly, but some of it is explained in Galbraith's book (see $\S15.7$ and $\S15.8$). These attacks do not apply to every curve over an extension field, but show that you'll have to be more careful.