The question is stated as follow:
By a prime field we mean a field which contains no proprer subfield(that is, no subfield other than itself). Find all prime fields. (Hint: Use the characteristic.)
In the previous question in the book that Im following, I proved that if the characteristic of a field $\frak{F}$ is $p$, the natural numbers $\mathbb{N}_\frak{F}$ of the field, determines a subfield of $\frak{F}$ which is contained in everyother subfield of $\frak{F}$.
From this I suppose its possible to show that the prime of field of any finite field with chracteristic $p$ is isomorphic to eachother, but It still unclear to me if prime fields related to fields with characteristic $0$ or not being a prime all are isomorphic in some way.
Any tips will be appreciated, another related question is, Starting from a finite field with characteristic $n$ where $n$ is not a prime, how can I guarantee that the prime field related to this field have characteristic $m$ where $m$ is a prime?