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Abstract Algebra

  1. Dec 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Let F be a field of characteristic p>0 and let E = F(a) where a is separable over F. Prove that E=F(a^p).


    2. Relevant equations



    3. The attempt at a solution
    I know that maybe show how mod F(a) = mod F(a^p) or something around there.
     
  2. jcsd
  3. Dec 3, 2009 #2
    So, since the characteristic is p, a prime E is generated by some field F adjoined some set of elements, and for each minimal polynomial associated to these elements there are no duplicate roots.

    Now, there are two cases to consider E is finite or E is infinite. If E is finite then you're done, since E is iso to Z mod p^k. If E is infinite (for example Z4(u) where u is some transcendental element) then you have some work to do. Consider what frobenius endomorphism
    http://en.wikipedia.org/wiki/Frobenius_endomorphism
    Tells about the minimal polynomial of a.
     
  4. Dec 3, 2009 #3

    Hurkyl

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    No it's not. Such an isomorphism can only exist when k=1 and E is a field whose cardinality is p.
     
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