Prove the uniqueness of the minimal polynomial

Question

I have here the definition:

Let $T$ be a linear operator on a finite-dimensional vector
space $V$ over the field $F$. The minimal polynomial for T is the (unique)
monic generator of the ideal of polynomials over $F$ which annihilate $T$.

I would like to know how to prove the uniqueness of it, how would I start?

Answer 1

You can start with proving

If $f(T)=g(T)=0$ for some polynomials $f,g$, then $\gcd(f,g)(T)=0$

and then the uniqueness will follow from a simple proof by contradiction.