1. ## 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!

2. Originally Posted by morganfor
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 $\displaystyle 0 \neq r \in R$ and $\displaystyle p(x)=x^n + a_1x^{n-1} + \cdots + a_{n-1}x+a_n \in F[x]$ be the minimal polynomial of $\displaystyle r.$ then $\displaystyle a_n \neq 0$ and thus: $\displaystyle r(-a_n^{-1}r^{n-1}-a_n^{-1}a_1r^{n-2} - \cdots - a_n^{-1}a_{n-1})=1. \ \Box$

3. 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?

4. Originally Posted by morganfor
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 $\displaystyle a_n$ over to the other side and dividing by $\displaystyle a_n$ both sides.