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Math Help - Prime Field

  1. #1
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    Prime Field

    I need the proof:

    The set of rational numbers (Q) is a Prime Field.
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  2. #2
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    Quote Originally Posted by manik View Post
    I need the proof:

    The set of rational numbers (Q) is a Prime Field.

    Let \mathbb{F} be any field with characteristic zero, then show that the map q\mapsto q\cdot 1_\mathbb{F}\,,\,\,q\in\mathbb{Q} , defines an injection (of rings) of \mathbb{Q}\,\,\,into\,\,\,\mathbb{F} .

    Another way: first prove that n\mapsto n\cdot 1_\mathbb{F} is an injection \mathbb{Z}\rightarrow \mathbb{F} , and then extend this to inverses within \mathbb{Q} by defining \frac{1}{n}\mapsto n^{-1}\in\mathbb{F},\,0\neq n\in\mathbb{Z}

    Finally , deduce that any field of char. zero contains an isomorphic copy of \mathbb{Q} ...

    Tonio
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