Question

There exists a field with..... elements

Answer 1

There exists a field with two elements. An example of such a field is $(\mathbb Z_2,+,\cdot )$ where $\mathbb Z_2=\{0,1\}$ and $0+0=1+1=0, \ 0+1=1+0=1,\ 0\cdot 0 = 0\cdot 1 = 1\cdot 0 =0,\ 1\cdot 1=1.$