Ring and Linear Transformation

[Treadstone 71]

Let $$g(x)\in F[x]$$, $$T\in L(V)$$. Let $$F[T]$$ be a ring generated by $$g(T)$$. Show that if $$g(T)$$ is invertible, then $$g^{-1}(T)\in F[T]$$.

No idea what do do.

 

[Hurkyl]

The statement of the problem is confusing. I'm going to assume V is a vector space over the field F.

Now, F[T] is not just "a" ring -- it is a very specific ring. It is (isomorphic to) the subring of L(V) that is generated by F and T.

So, I'm confused when you say that you let it be "a ring generated by g(T)".

Did you mean to say something to the effect if:

"Suppose g(T) generates F[T]"

or maybe

"Let F[g(T)] be the ring generated by g(T)"

or even just

"So g(T) is an element of F[T]"

?


(I suspect you meant the first one -- work out what that really means)

 

[Treadstone 71]

It's either the first or third. Let's assume the first, since I must show that the inverse of g(T) is a positive powered polynomial.

 

[Hurkyl]

Okay, so what does it mean that g(T) generates F[T]? Any interesting particular cases?