Prove there is no field of order 4

Question

I've begun a solution by finding the orders of the elements and finding the relations between them hoping to come to some contradiction.

However I've come across this question, whcih seems to suggest there is a field of order 4? However, I haven't yet covered field extensions. Without field extensions is there no field of order 4?

Answer 1

Consider the set $F=\{0,1,a,a+1\}$. Define in it the operations $+$ and $\times$ in the obvious way with the relations $$x+x=0\;\forall x\in F$$ and $$a^2=a+1$$ and you have the field.

Answer 2

There always exists a field of order $p^n$. Consider the roots of the equation $x^{p^n} -x=0$. The roots of the equation form a field of order $p^n$. You can clearly see that it has $p^n$ roots from fundamental theorem.and all are distinct.