1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

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


    User Avatar
    Science Advisor
    Homework Helper

    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]
    [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].
    [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


    User Avatar
    Science Advisor
    Homework Helper

    Much better. You're welcome!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook