Prime subfield

Prime subfield: Let $K$ be a field. The subfield of $K$ generated by $1$ is isomorphic to $\mathbb Q$ if $\operatorname{char}(K)=0$, and is isomorphic to $\mathbb F_p$ if $\operatorname{char}(K)=p>0$.

Apply the characteristic dichotomy for domains to the integral domain underlying a field to see that $\operatorname{char}(K)$ is $0$ or a prime $p$. The characteristic then controls the image of $\mathbb Z\to K$; in characteristic $0$ one takes the induced fraction field to obtain a copy of $\mathbb Q$, while in characteristic $p$ the image is $\mathbb F_p$.