# Subfield of the field of complex numbers

1. Feb 6, 2010

### cocobaby

Can anyone prove to me why each subfield of the field of complex numbers contains every rational numers?

Last edited: Feb 6, 2010
2. Feb 6, 2010

### Hurkyl

Staff Emeritus
Hrm. Let's start with a simpler question: can you say why they must all contain the number 2? What other numbers can you say something about?

3. Feb 6, 2010

### HallsofIvy

Staff Emeritus
Any subfield of the complex numbers must include 0 and 1. Since it is closed under addition, it must include every positive integer. Since it has additive inverses, it must include every negative integer. Since every member except 0 has a multiplicative inverse, ....