I need the proof:

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

Printable View

- Apr 12th 2010, 03:00 AMmanikPrime Field
I need the proof:

The set of rational numbers (Q) is a Prime Field. - Apr 12th 2010, 03:46 AMtonio

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

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

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

Tonio