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Math Help - Isomorphism of fields

  1. #1
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    Isomorphism of fields

    Let φ: F1 --> F2 be an isomorphism field. Prove that φ(1) = 1. (That is, prove that φ must map the multiplicative identity of F1 to the multiplicative identity of F2.)

    I know that there is a "1" element in F1 and a "1" element in F2, because they're fields. I also know that because φ is isomorphic, there exists a bijection from F1 to F2.
    I need to prove that φ takes the "1" element in F1 to the "1" element in F2.

    Can anyone help me from here? Thanks in advance.
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  2. #2
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    Quote Originally Posted by page929 View Post
    Let φ: F1 --> F2 be an isomorphism field. Prove that φ(1) = 1. (That is, prove that φ must map the multiplicative identity of F1 to the multiplicative identity of F2.)

    I know that there is a "1" element in F1 and a "1" element in F2, because they're fields. I also know that because φ is isomorphic, there exists a bijection from F1 to F2.
    I need to prove that φ takes the "1" element in F1 to the "1" element in F2.

    Can anyone help me from here? Thanks in advance.


    As we've an isomorphism, \forall b\in F_2\,\exists a\in F_1\,\,s.t.\,\,\phi(a)=b\,,\,\,1_2=\phi(a') , but then

    b=1_2\cdot b = \phi(a')\phi(a)=\phi(a'a) . Now use that the above a is unique (because of 1-1) and...

    Tonio
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