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Thread: Field

  1. #1
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    Field

    I need the proof:

    The set of Rational numbers has no proper sub-field.
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  2. #2
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    Q is a prime field of characteristic 0, i.e. Q can be embedded in any field of characteristic 0.
    But subfield preserves characteristic.
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  3. #3
    Junior Member nimon's Avatar
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    Alternatively, if $\displaystyle F\subseteq\mathbb{Q}$ is a subfield then $\displaystyle 1\in F$, so that $\displaystyle 1+1,1+1+1,\ldots\in F$ so $\displaystyle \mathbb{N}\subseteq F$. But then $\displaystyle -1,-1-1,\ldots \in F$ so that $\displaystyle \mathbb{Z}\subseteq F$. But for each $\displaystyle 0\neq z\in\mathbb{Z}$ we have that $\displaystyle z^{-1} \in F$, so that $\displaystyle pq^{-1} \in F$ for any $\displaystyle p,q\in\mathbb{Z},q\neq 0$. Hence $\displaystyle \mathbb{Q}\subseteq F.$
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