1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fields and Subfields

  1. Dec 29, 2004 #1
    I am self studying linear algebra from `Linear Algebra' by Hoffman and Kunze.
    One of exercise Q is:
    Prove that Every subfield F of C contains all rational numbers.

    But doesn't the set {0,1}(with the usual +,-,.) satisfy all conditions to be a field?
     
    Last edited: Dec 29, 2004
  2. jcsd
  3. Dec 29, 2004 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    what's 1+1?
     
  4. Dec 29, 2004 #3

    Janitor

    User Avatar
    Science Advisor

    As Matt implies, closure under the operations is a requirement.
     
  5. Dec 29, 2004 #4
    EEK!I forgot about 1+1 :(
    to have closure under addn & subtr you need to have Z.
    to have closure under multiplication and division(or existance of x^-1 for all x) you need Q.Therefore All subfields of C should have atleast Q in them.
    Is my proof correct?
     
  6. Dec 29, 2004 #5
    Wait a minute!
    My set can be a field with characteristic 2 (1+1=0).(or is it characteristic 1)
    Which brings me to the next Question.
    P.T. All zero characteristic fields contain Q.
    Any hints how to begin?
    Thanks is advance
     
  7. Dec 29, 2004 #6

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    Yes that's basically correct:

    1 and 0 must be elements (actually Im a little unclera on this is the trivial field technically a subfield of C?) thus any 1+1+1...+1 is also an element so all the natural nunmbers must be elements and by additve inverse all integers must be elements. Any number in Q can be given by n*1/m where n and m are integers (m not equal to zero), by muplicative inverse 1/m must be in the any subfield of C, therefore any subfield of C has Q as a subfield.
     
  8. Dec 29, 2004 #7
    Thanks .but is the charactesitic 1 or 2?
     
  9. Dec 29, 2004 #8
    Please dont give away the whole ans.Just gimme a hint.Thanks anyway
     
  10. Dec 29, 2004 #9

    jcsd

    User Avatar
    Science Advisor
    Gold Member

    Yes, but it's not a subfield of C though is it.

    Just look at the definition of a field with charestic 0.
     
  11. Dec 30, 2004 #10
    True
    Why?
     
  12. Dec 30, 2004 #11

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    because of the prefix sub. If it is a subfield then adding two elements in the subfield must give the same answer as adding them in the field, so if 1+1..+1=0 in the subfield, it equals zero in the field and hence the field has characteristic p for soem prime.


    All fields must contain 0 and 1 and these are distinct (so the set {0} with addition and multiplication isn't a field, jcsd), so all fields of char 0 contain a copy of Q. The proof is the same as for the large field being C. You didn't actually use anything other than it was a field of characteristic zero did you?
     
    Last edited: Dec 30, 2004
  13. Dec 30, 2004 #12
    So {0,1,+,.} is a field with charecteristic 2.But it is not a subfield of C.
    Thanks.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Fields and Subfields
  1. Subfields of a field (Replies: 2)

  2. Subrings and Subfields (Replies: 14)

  3. Isomorphic Subfields (Replies: 1)

Loading...