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Automorphisms and some maps that are bijective

  1. Jan 23, 2007 #1
    I am beginning abstract algebra and am having difficulty showing that some maps are bijective. It is a function f:V -> V where:

    f_1 : ( 1 -> 1 and a -> a) and...
    f_2 : ( 1 -> 1 and a -> a') where a and a' are the zeros of a polynomial.

    f_1 seems trivial. Whatever you plug in you get again so I can not see how it could not be bijective. Going back to my topology class it is the identity map I believe. I am having difficulties getting f_2 where a' is the algebraic conjugate. Does anyone have suggestions about how to think about it. I just can not seem to get past the first hurdle of where to start.

    Sorry for the formatting. I do not know how to make piecewise functions in tex. Thanks for any help in advance.
  2. jcsd
  3. Jan 23, 2007 #2


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    I'm guessing V is the simple field extension of some field F generated by some a with minimal polynomial f(x), and a' is another root of f(x) which lies in V. Is this right? Please include this information next time.

    If you've shown f_1 is the identity, you've shown its a bijection. Are you having trouble seeing why the identity is a bijection?

    Have you been able to show what the elements in V look like in terms of a? I'm guessing you have if you've done the first part. If so, show there's a similar representation in terms of a', and use this to show f_2 has a two sided inverse, and therefore is a bijection.
  4. Jan 23, 2007 #3
    Yes, I did understand the identity map. And your explanation of the two sided inverse makes sense. Thanks. I understand it.
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