Construct a field of order 25.
I don't really understand this question, am I suppose to list the polynomials of degree 5?
The question is saying to explicity create a field, we know a field of 25 elements exists but we want to explicitly state it.
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 $\displaystyle p(x)$ be a (nonconstant) polynomial over a field $\displaystyle F$. Then $\displaystyle p(x)$ is irreducible over $\displaystyle F$ if and only if $\displaystyle F[x]/\left< p(x) \right>$ is a field.
Consider $\displaystyle F = \mathbb{Z}_5$, and the polynomial $\displaystyle p(x) = x^2 + 2$ 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 $\displaystyle F[x]/\left< x^2+2 \right>$ is a field. How many elements are in this field? Note any element in this field can be written as $\displaystyle ax+b$ since of uniquneness $\displaystyle a$ and $\displaystyle b$ can be any elements of the field $\displaystyle F$ since there are five elements in total we can write $\displaystyle 5\cdot 5 = 25$.