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Math Help - Subfield Q

  1. #1
    Senior Member sfspitfire23's Avatar
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    Subfield Q

    Show that K is a subfield of F if and only if 0\neq a\in K implies that a^{-1}\in K

    So, we know that all fields have inverses. So, if 0\neq a then we can say a=n\in K. Then we have n+(-n)=0 and nn^{-1}=1=n^{-1}n because a\neq 0. Now this doesn't completely show \rightarrow part of the iff does it? If we know that the element has an inverse, can we say it satisfies all of the other axioms of a field?
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  2. #2
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    Show that K is a subfield of F if and only if implies that
    Arn't there other hypotheses? I mean, if that equivalence was true, then (\{0,1\},+,\times ) would be a subfield of (\mathbb{Q},+,\times ), and this is obviously wrong.

    A subfield is a subring, so it is an additive subgroup.
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  3. #3
    Senior Member sfspitfire23's Avatar
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    no other hypotheses, just what i had written. Interesting. But why is 0 in the first ring when a cannot equal 0?
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  4. #4
    Senior Member sfspitfire23's Avatar
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    Ah, i believe this proof is quite trivial. Since a subfield is a subgring, all we need to say is that in additionto the things required to be a subfield we only need to show it also has an inverse.
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