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Math Help - algebraic extension proof

  1. #1
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    algebraic extension proof

    I need help with this proof. Let E be an algebraic extension of field F. If R is a ring and F is contained in R is contained in F, show that R must be a field.
    Thanks!
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  2. #2
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    Quote Originally Posted by morganfor View Post
    I need help with this proof. Let E be an algebraic extension of field F. If R is a ring and F is contained in R is contained in E, show that R must be a field.
    Thanks!
    let 0 \neq r \in R and p(x)=x^n + a_1x^{n-1} + \cdots + a_{n-1}x+a_n \in F[x] be the minimal polynomial of r. then a_n \neq 0 and thus: r(-a_n^{-1}r^{n-1}-a_n^{-1}a_1r^{n-2} - \cdots - a_n^{-1}a_{n-1})=1. \ \Box
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  3. #3
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    Thanks! I get it all the way up until the last part. How did you get the final polynomial and how is it that when you multiply it by r you get 1?
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  4. #4
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    Quote Originally Posted by morganfor View Post
    Thanks! I get it all the way up until the last part. How did you get the final polynomial and how is it that when you multiply it by r you get 1?
    By moving a_n over to the other side and dividing by a_n both sides.
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