# Show that a field of characteristic $0$ is infinite

Show that if a field $F$ has characteristic $0$, then it is of infinite order.

Please someone help me. It is an unsolved exercise in my book, it is not homework.

• If $n\times 1\neq 0$ for every $n$ then these elements are all different – user8268 Sep 15 '13 at 13:51
• Well, suppose that F has finitely many elements. Then the subset {1, 1+1, 1+1+1,...} of F has only finitely many distinct elements. What would that imply? – Keshav Srinivasan Sep 15 '13 at 13:52
• If it were finite, adding the multiplicative identity $1$ a number of times (say $p$) gets you the additive identity $0$. – mrk Sep 15 '13 at 13:55

## 2 Answers

A ring $R$ has characteristic zero if there is a monomorphism $\phi : \mathbb{Z} \to R$. What does this imply about the relationship between $|\mathbb{Z}|$ and $|R|$?

• Surely you meant to write $|R|$ at the end of your answer. – Asaf Karagila Sep 15 '13 at 14:02
• Indeed. But $\mathbb{R}$ also has characteristic zero... – Elchanan Solomon Sep 15 '13 at 14:18
• Beautiful solution. I can go to bed happy. – Daniel Montealegre Sep 18 '13 at 4:53

A field with characteristic zero contains a subfield isomorphic to Q. The results follows by Q