Let , and let be an extension of of degree . Then is unique among all degree- extensions. To see why, note that , so is the splitting field for over . As splitting fields are unique, the result follows.
Conversely, for every , there exists a field with . To prove that, it’s enough to prove that the splitting field of has exactly elements. The polynomial has a nonzero constant, so it doesn’t have zero as a root.
If the field has fewer than nonzero elements, then it must have repeated roots. But its derivative is , whose sole irreducible factor, , doesn’t divide , contradicting the result that the polynomial has a repeated root.
To see that it doesn’t have more than elements, note that the roots of the polynomial together with 0 form a field. This is because if and , then clearly, and by repeatedly applying the Frobenius automorphism.
That completes the classification of finite fields, which states that there is a unique finite field for each prime power order, and no finite field with an order that isn’t a prime power.
On another note, be nice to me. If you ask politely, I might go back to my earlier math posts and edit them to incorporate .