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If K is contained in R contained in F and F is algebraic over K, show R is a field

  1. Feb 18, 2012 #1
    Let [itex] K \subseteq F [/itex] be fields and let R be a ring such that [itex] K \subseteq R \subseteq F [/itex]. If F is algebraic over K, show that R is a field.

    If F is algebraic over K, then every element of F is a root of some polynomial over K[x]. But since K is contained in R, every element of F is thus a root of some polynomial over R[x]. I want to show that every nonzero element of R has an inverse which would show that it is a field. The elements in K are obviously invertible so I need to show that any element in R that is not in K is also invertible. I am having trouble with this part and can't think of a way to show this. Can someone offer a hint or two in the right direction?

    Help is greatly appreciated.
     
  2. jcsd
  3. Feb 19, 2012 #2

    morphism

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    Re: If K is contained in R contained in F and F is algebraic over K, show R is a fiel

    Let a be a nonzero element of R. Here are two independent (but ultimately equivalent) hints to help you show that the inverse of a belongs to R.

    Hint A: K(a)=K[a].

    Hint B: Write down the inverse of a in F, using the fact that a is algebraic/K.
     
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