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Math Help - Subfield Isomorphic To Q

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    Senior Member vincisonfire's Avatar
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    Subfield Isomorphic To Q

    I would like to know how to show that if the characteristic of a field is 0 then it has a subfield isomorphic to \mathbb Q .

    I defined an homomorphism  f : \mathbb F \rightarrow \mathbb Q
     f(n) = n \cdot 1
    I found,  Ker(f) = \{0\} The map is thus injective.
    I thought I might find a surjective map to \mathbb F/\{0\} and use the first isomorphism theorem to conclude that  \mathbb F/\{0\} is a subfield isomorphic to  \mathbb Q .
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    Quote Originally Posted by vincisonfire View Post
    I would like to know how to show that if the characteristic of a field is 0 then it has a subfield isomorphic to \mathbb Q .

    I defined an homomorphism  f : \mathbb F \rightarrow \mathbb Q
     f(n) = n \cdot 1
    I found,  Ker(f) = \{0\} The map is thus injective.
    I thought I might find a surjective map to \mathbb F/\{0\} and use the first isomorphism theorem to conclude that  \mathbb F/\{0\} is a subfield isomorphic to  \mathbb Q .
    Define an embedding f: \mathbb{Z} \to F by f(n) = n\cdot 1. This notation means n\cdot 1 = 1 + ... + 1, n times if n>0 and n\cdot 1 = - (1+...+1) if n<0 and 0\cdot 1 = 0. Therefore, \mathbb{Z} is embedded in F, however if F contains by an embedding \mathbb{Z} it must contain by an embedding a field of quotients of \mathbb{Z} i.e. \mathbb{Q}.
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