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Show F is isomorphic to F/{0}

  1. May 27, 2015 #1
    1. The problem statement, all variables and given/known data
    Let F be a field. Show that F is isomorphic to F/{0}
    2. Relevant equations


    3. The attempt at a solution
    By the first ring isomorphic theorem, kernel of the homomorphism is an ideal which is either {0} or I. Hence F isomorphic to F/{0}

    I think I misunderstood the problem can anyone check my work where I did wrong
     
  2. jcsd
  3. May 27, 2015 #2

    Zondrina

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    For two fields to be isomorphic, there has to be a bijective homomorphism between them:

    $$\phi : F \rightarrow (F - \{0\})$$

    You need to define the map you used to show ##\text{ker}(\phi)## is an ideal of ##F##. You then need to show ##\ker(\phi)## really is an ideal.

    Then I would be convinced.

    You could also go the long way by proving the map you've defined is a bijective homomorphism between the fields, which would imply the fields are isomorphic to each other.
     
  4. May 28, 2015 #3
    Suppose $\phi : F\rightarrow F$ is an identity homomorphism, then $\ker(\phi)=\{0\}$ is an ideal of $F$ and hence $F/\{0\}\cong F$ by the first ring isomorphism.
     
  5. May 29, 2015 #4

    Zondrina

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    Are you saying ##\phi(f) = f, \forall f \in F##? If so there is a problem because ##\phi(0) = 0## and ##0 \notin (F - \{ 0 \})##.
     
  6. May 29, 2015 #5

    micromass

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    The OP means ##F/\{0\}##, which means the ring ##F## modulo ##\{0\}##. He does not mean the set theoretic difference ##F\setminus \{0\}## (in that case, the result in the OP is obviously false).
     
  7. May 29, 2015 #6

    micromass

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    Correct.
     
  8. May 30, 2015 #7

    Zondrina

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    Oh I thought the OP was intending to mean the set theoretical difference.

    I never knew \setminus was the way to do that, so I guess I learned something too.

    The OP is correct if they were intending modulo, but I definitely would have been a bit more explicit when asking.
     
  9. May 30, 2015 #8

    micromass

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    I agree, I was very confused too.
     
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