Minimal polynomials and invertibility

[Treadstone 71]

Let T\in L(V). Let g(x)\in F[x] and let m(x) be the minimal polynomial of T. Show that g(T) is invertible \Leftrightarrow \gcd (m(x),g(x))=1.

Backwards is easy. For forwards, suppose I say that g(T) is invertible implies that g(T)(v)=0 \Rightarrow v=0 and therefore g(x) prime, therefore it is not divisible, and therefore \gcd (g,m)=1. Is that correct?

Why is latex not showing up?

 

[topsquark]

Let T\in L(V). Let g(x)\in F[x] and let m(x) be the minimal polynomial of T. Show that g(T) is invertible \Leftrightarrow \gcd (m(x),g(x))=1.

Backwards is easy. For forwards, suppose I say that g(T) is invertible implies that g(T)(v)=0 \Rightarrow v=0 and therefore g(x) prime, therefore it is not divisible, and therefore \gcd (g,m)=1. Is that correct?

Why is latex not showing up?

They had some sort of server disaster yesterday, I take it.

-Dan