The question is list all those integers $n$ such that $1 \leq n \leq 10$ and there exists a field with $n$ elements.
Of course $n$ can take values $2,3,5,7$ since they are prime and we know $ \mathbb Z_p$ is a field iff $p$ is a prime.
Next I will use the concept that if $p(x)$ is a irreducible polynomial then the ideal $<p(x)>$ is maximal, and if $I$ is a maximal ideal of a ring $R$ then $R/I$ is a field.
Now I consider the ideal $<x^2+x+1>$ over $\mathbb F_2(x)$, since $x^2+x+1$ is irreducible in $\mathbb F_2(x)$ then $\mathbb F_2(x)/<x^2+x+1>$ is a field. Any element of this field will have a form $ax+b$, where $(a,b) \in \mathbb F_2$, this implies this field has $4$ elements.
By the same type of argument, I have found the fields $\mathbb F_2(x)/<x^3+x^2+x+1>$ and $\mathbb F_3(x)/<x^2+x+1>$ have $8$ and $9$ elements respectively.
However I cannot find any field containing $6$ or $10$ elements. How to confirm whether there really are fields containing $6$ or $10$ elements?
Now I found that there are fields with $2,3,4,5,7,8,9$ elements. My method is a bit tedious. Are there any simple and direct computation to find so?