Subfield of the field of complex numbers

[cocobaby]

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

 

[Hurkyl]

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?

 

[HallsofIvy]

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, ...