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Math Help - Field

  1. #1
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    Field

    Let R be an integral domain having a finite number of elements. Prove R is a field.

    Hint: Given any nonzero real number a, consider the list of elements ab for all real numbers b.
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  2. #2
    Junior Member
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    First of all, the "numbers" a and b are not necessarily real numbers: they're elements of an arbitrary integral domain R.

    So, pick a\not =0, and consider the function

    \phi :R\to R

    given by \phi (r)=a\cdot r. Show that this function is injective (you will need to use the integral domain hypothesis), argue that it must thus also be surjective, and conclude.
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