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Characteristic Zero

  1. Sep 16, 2009 #1
    Can anyone explain to me why each field of characteristic zero contains a copy of the rationals, or a subfield that's isomorphic...
  2. jcsd
  3. Sep 16, 2009 #2


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    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
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