What is the difference between a field a subfield

  1. For example, my notes say, "Q (rationals) is a subfield of R (reals). Z (integers) is not a subfield of R. Any subfield (together with the addition and multiplication) is again a field".

    This just doesn't make any sense to me.

    Oops, this was suppose to be in the homework section - sorry.
     
  2. jcsd
  3. lurflurf

    lurflurf 2,325
    Homework Helper

    That should say something like
    "A subfield of a field is any subset of the field that is itself a field (with the same operations)."
    What you have
    "Any subfield (together with the addition and multiplication) is again a field".
    Is true, but not very useful without context.
     
  4. I still don't understand why Q is a subfield of R, but Z isn't.
     
  5. Is Z a field?
    What are the field axioms?
     
  6. Oh, is it not a field because division of 2 integers can produce a number that isn't an integer?
     
  7. lurflurf

    lurflurf 2,325
    Homework Helper

    ^Yes. A subset is a subfield if it is itself a field (with the same operations). Z is not a field, so it is not a subfield.
     
  8. Thank-you everyone!
     
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