Thread: algebraic extension proof
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Old December 4th, 2008, 01:19 AM
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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.
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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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