I seem to hold a very loose grasp of the concept of fields - I've encountered this question: Show that every finite field with characteristic $p$ contains $\mathbb{Z}_p$ (i.e. $\mathbb{Z}_p = \{0,...,p-1\}$). Now my question is what exactly is a field of characteristic $p$?
My second question: Show that every field of characteristic $0$ contains $\mathbb{Q}$ as a subfield?
Regards.
If $F$ is a field use the ring morphism $h:\Bbb Z\to F$ defined by $n\mapsto n.1_F$ and regarded it' kernel. It's kernel is a subgroup of $(\Bbb Z,+)$ so it is of the form $\ker h=p\Bbb Z$. Now discuss the cases $p=0$ and $p\neq 0$.