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Math Help - Integral domain question

  1. #1
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    Integral domain question

    Let R,S be integral domains of characteristic zero. Let K be the minimal field extension containing \text{Fr}(R) and \text{Fr}(S) as subfields. Does there exist an integral domain T such that K = \text{Fr}(T) with R and S are both subrings of T?
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  2. #2
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    Re: Integral domain question

    Well, sure, but it's probably not the answer you're looking for: clearly $K$ fits the bill.

    What you are probably looking for is the sub-ring generated by $R$ and $S$ (I think you can characterize this ring as $\{r+s+r's':r,r' \in R, s,s' \in S\}$). Since this ring is contained in $K$ it has no zero-divisors, and is thus an integral domain. Since its field of fractions is contained in any field that contains it, $\text{Fr}(\langle R\cup S\rangle) \subseteq K$.

    Since this field of fractions also contains $R$ and $S$, it must also contain $\text{Fr}(R),\text{Fr}(S)$, and by the minimality of $K$, it must BE $K$.
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  3. #3
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    Re: Integral domain question

    Excellent! Thank you! I had been looking through old notes from my Ring Theory class, and remembering my professor writing so fast, I couldn't pay attention to him and copy what he was writing at the same time. I feel like I learned so little in that class. One of these years, I want to go back and try to relearn it all.
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