1. ## Prime Field

I need the proof:

The set of rational numbers (Q) is a Prime Field.

2. Originally Posted by manik
I need the proof:

The set of rational numbers (Q) is a Prime Field.

Let $\mathbb{F}$ be any field with characteristic zero, then show that the map $q\mapsto q\cdot 1_\mathbb{F}\,,\,\,q\in\mathbb{Q}$ , defines an injection (of rings) of $\mathbb{Q}\,\,\,into\,\,\,\mathbb{F}$ .

Another way: first prove that $n\mapsto n\cdot 1_\mathbb{F}$ is an injection $\mathbb{Z}\rightarrow \mathbb{F}$ , and then extend this to inverses within $\mathbb{Q}$ by defining $\frac{1}{n}\mapsto n^{-1}\in\mathbb{F},\,0\neq n\in\mathbb{Z}$

Finally , deduce that any field of char. zero contains an isomorphic copy of $\mathbb{Q}$ ...

Tonio