# Prove that every non-zero vector in V is a maximal vector

1. Nov 19, 2012

### Ninty64

1. The problem statement, all variables and given/known data
Let V be a finite dimensional vector space and T is an operator on V. Assume $μ_{T}(x)$ is an irreducible polynomial. Prove that every non-zero vector in V is a maximal vector.

2. Relevant equations
$μ_{T}(x)$ is the minimal polynomial on V with respect to T.

3. The attempt at a solution
I am not sure how to go about solving this problem. I know that if T is cyclic then $\exists \vec{v} \in V$ such that $<T,\vec{v}>=V$, then for any $\vec{w}\in V$,
$\exists f(x) \in F[x]$ such that $\vec{w} = f(T)(\vec{v})$
and it follows that $μ_{T,\vec{w}}(x) = \frac{μ_{T}(x)}{gcd(μ_{T}(x),f(x))}$
However, since $μ_{T}(x)$ is an irreducible polynomial, then $gcd(μ_{T}(x),f(x)) = μ_{T}(x)$ or $gcd(μ_{T}(x),f(x)) = 1$
If $gcd(μ_{T}(x),f(x)) = μ_{T}(x)$, then $μ_{T,\vec{w}}(x) = 1$, which implies that $\vec{w} = I(\vec{w}) = \vec{0}$
If $gcd(μ_{T}(x),f(x)) = 1$, then $μ_{T,\vec{w}}(x) = μ_{T}(x)$, which means that $\vec{w}$ is a maximal vector. Thus every non-zero vector in V is a maximal vector.

However, I'm not sure how to prove that T is cyclic.

I also attempted to find the minimal polynomials with respect to the basis vectors for V.
Since $μ_{T}(x) = lcm(μ_{T,\vec{v_1}}(x),μ_{T\vec{v_2}}(x),...,μ_{T,\vec{v_n}}(x))$
$\Rightarrow μ_{T}(x)=μ_{T,\vec{v_1}}(x)=μ_{T,\vec{v_2}}(x)=...=μ_{T,\vec{v_n}}(x)$ since the minimal polynomial is irreducible. But then I didn't know where to go from there either.

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