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Math Help - Fields

  1. #1
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    Fields

    Which subfields of C are closed under addition, subtraction, multiplication and division, but fail to contain 1.
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    All subfields of \mathbb C must contain 1. Or do you mean subrings of \mathbb C\,?

    The only subring of \mathbb C closed under division and not containing 1 is the trivial subring \{0\}. (Note that division is not permitted in this ring and so it is vacuously closed under division.) For any other subring R, if a is a nonzero element, then the fact that R is closed under division means that 1=\frac aa\in R.
    Last edited by TheAbstractionist; June 14th 2009 at 08:32 AM.
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  3. #3
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    oh sorry, i meant subsets!
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  4. #4
    Senior Member TheAbstractionist's Avatar
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    Well, vacuously speaking, the empty set \O would satisfy the given conditions. Presumably, you want a nonempty subset. Any nonempty subset of a field that is closed under addition, subtraction and multiplication is a subring, so what I said in my previous post applies.
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