I have to ways to to construct such a field I will do it with polynomials since that is what you mentioned and I assume that is what you want.
First we need to know a theorem: Let be a (nonconstant) polynomial over a field . Then is irreducible over if and only if is a field.
Consider , and the polynomial is irreducible since it is of degree two and it has no zero (just check for zeros x=0,1,2,3,4).
Now since it is irreducible by the theorem is a field. How many elements are in this field? Note any element in this field can be written as since of uniquneness and can be any elements of the field since there are five elements in total we can write .