Characteristic Zero: Explaining Rationals & Subfield Isomorphism

[CoachZ]

Can anyone explain to me why each field of characteristic zero contains a copy of the rationals, or a subfield that's isomorphic...

 

[Office_Shredder]

If you have a field with characteristic zero.

1 is in the field. Also, 1+1 is in the field. And 1+1+1, etc. Let's call 1+1 2 for now, 1+1+1 3 for now. Since the characteristic is zero, none of these are zero, so the map 1+1+...+1 k times can be mapped to k, and -1-1-1...-1 k times can be mapped to -k and we have a copy of the integers inside the field.

Not only do we have the integers, but since it's a field, we also have 1/2, 1/3, etc. anything of the form 1/m for m an integer (non-zero) and since it's closed under multiplication, we have everything of the form n*(1/m) where n and m are integers in the field (so elements of the form 1+1+1... or -1-1-1...) The map n/m (the rational number) to n*(1/m) (the product in the field) gives you that the subfield of all things of the form n*(1/m) is in fact isomorphic to the rationals