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Do field homomorphisms preserve characteristic

  1. Jun 28, 2009 #1
    1. The problem statement, all variables and given/known data

    Given two fields F,E with different characteristic. Prove or disprove the following statement: "Field homomorphisms between fields of different characteristic cannot exist"

    2. Relevant equations
    T : F1 --> F2 is a field homomorphism if
    1) T(a+b) = T(a) + T(b)
    2) T(ab) = T(a)T(b)
    3) T(1) = 1
    4) T(0) = 0.


    3. The attempt at a solution
    Intuition says no...

    All field homorphisms are injective. So T:F --> E where F has bigger order than E cannot exist. On the other hand, if E has bigger order than F, F must contain an isomorphic copy of E.

    Hmm, not sure where to go from here. Here is my attempt... Suppose we do have a hom from F to E where char E is bigger than char F. Then by the fundamental homomorphism theorem, F/kerT is isomomorphic to E. However since T is injective the kernel is trivial. Therefore F is isomorphic to E contradicting the assumption of different characteristic...
     
  2. jcsd
  3. Jun 28, 2009 #2

    Dick

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    That is nonsense. Look up the definition of 'field characteristic'. Read it several times. Then look at your requirement T(1)=1.
     
  4. Jul 6, 2009 #3
    Right read the definitions

    [tex]T(1_{F1}) = 1_{F2}[/tex]
    Thus,
    [tex]T(n1_{F1}) = nT(1_{F1}) = n1_{F2}[/tex]
    Thus suppose char(F1) = m,
    [tex]T(m1_{F1}) = T(0) = 0 = mT(1_{F1}) = m1_{F2}[/tex]
    Therefore, char(F2) <= m.
    Suppose p < m satisfies [tex]p1_{F2} = 0[/tex].
    Then,
    [tex]T^{-1}(p1_{F2}) = T(0) = 0 = T^{-1}(p1_{F2}) = pT^{-1}(1_{F2}) = p1_{F1}[/tex]
    Contradicting the minimality of m. Therefore, char(F2) = m.
     
  5. Jul 6, 2009 #4
    P.s. thank you!
     
  6. Jul 6, 2009 #5

    Dick

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    Much better. You're welcome!
     
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