/sci/ - Science & Math

p14 ex11

Let [math]0\neq a\in R[/math]. We need to show [math]a^{-1}\in R[/math]. Since [math]K[/math] is algebraic over [math]F[/math], there is a minimal polynomial over [math]F[/math], [math]p(x)=x^n+b_{n-1}x^{n-1}+...+b_0[/math] such that [math]p(a^{-1})=0[/math]. Rearranging and multiplying the latter equation by [math]a^{n-1}[/math] gives us [math]a^{-1}=-(b_{n-1}+...+b_0 a^{n-1})[/math], an equation that exists over [math]R[/math].