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Math Help - subfield proof

  1. #1
    ux0
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    subfield proof

    If R is a domain (integral domain), prove that there is no subfield K of Frac(R) such that

    <br />
R\subseteq K \subset Frac(R)

    Note: in the book the part that K is a subset of Frac(R) they used some funny notation, that basically looks like subset or equal to, but the line that means equal to was crossed out.... I just assumed that they mean the subset, which doesn't equal K.... but I'm not sure
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    Quote Originally Posted by ux0 View Post
    If R is a domain (integral domain), prove that there is no subfield K of Frac(R) such that

    <br />
R\subseteq K \subset Frac(R)

    Note: in the book the part that K is a subset of Frac(R) they used some funny notation, that basically looks like subset or equal to, but the line that means equal to was crossed out.... I just assumed that they mean the subset, which doesn't equal K.... but I'm not sure
    if r,s \in R, \ s \neq 0, then, since K is a field and it contains R, we have s^{-1} \in K. thus rs^{-1} \in K. but we know that Frac(R) is exactly the set of rs^{-1} with r,s \in R, s \neq 0. so K = Frac(R).
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