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Math Help - Z is not a field.

  1. #1
    No one in Particular VonNemo19's Avatar
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    Z is not a field.

    Why is \mathbb{Z} not a field?
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  2. #2
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    Quote Originally Posted by VonNemo19 View Post
    Why is \mathbb{Z} not a field?
    A field requires that every nonzero element has an inverse. The only invertable element of the integers are plus or minus one.

    For example 2 is not invertible in \mathbb{Z}

    2x=1 does not have any integer solutions as \frac{1}{2} \notin \mathbb{Z}
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Oh, OK. I was looking and looking at the definition of a field and comparing it to the properties of the integers and I couldn't find the missing ingredient. So, this little property of the integers, namely that of not having a multiplicitive inverse, is the only condition of the definition of a field that is not satitsfied, correct?
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Yes.
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