# Irreducible characteristic polynomial of a linear transformation

I am trying to understand why the linear transformation $T$ corresponding to the companion matrix of the minimal polynomial of an irreducible polynomial over $\mathbb{Q}[x]$ (for instance $x^3-x-1$) has no non-trivial $T$-invariant subspaces.

I know the minimal polynomial of $T$ is equal to the characteristic polynomial of $T$ and furthermore $T:\mathbb{Q}^4\rightarrow \mathbb{Q}^4$ has a cyclic vector $\alpha$ ( that is $\exists \alpha \in \mathbb{Q}^4$ such that $\mathbb{Q}^4 = \{ g(T)\alpha : g \in \mathbb{Q}[x]\}$.

Question: I have not been able to find a specific theorem in my textbook that tells you that when the minimal polynomial is irreducible then it has no non trivial T-invariant subspaces. My question is does there exist such a result or can we make a similar statement when $\mathbb{Q}$ is replaced by any field that is not algebraically closed?

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If $V$ is a $T$-invariant subspace, then the minimal polynomial of the endomorphism of $V$ induced by $T$ divides the minimal polynomial of $T$. [By the way, there is a typo in the title.] –  Pierre-Yves Gaillard Aug 10 '11 at 8:33
Thank you. So the existence of a non-trivial $T$ invariant subspace is equivalent to the existence of a root for the characteristic polynomial existing in the base field? –  user7980 Aug 10 '11 at 8:56
You’re welcome. The existence of a non-trivial T invariant subspace is equivalent to the existence of a non-trivial factor of the characteristic polynomial existing in the base field. [I think I can make this (slightly) more precise if you want.] –  Pierre-Yves Gaillard Aug 10 '11 at 9:02
Let $f$ be in $K[X]$, where $K$ is a field and $X$ an indeterminate. The ideals of the ring $K[X]/(f)$ correspond to the ideals of $K[X]$ which contain $f$, and thus to the factors of $f$. –  Pierre-Yves Gaillard Aug 10 '11 at 9:16
@Andrea: We're talking about companion matrices. (The minimal and characteristic polynomials must coincide.) (Look at the beginning of the question.) –  Pierre-Yves Gaillard Aug 10 '11 at 9:47

Let $K$ be a field, $X$ an indeterminate, $f\in K[X]$ a degree $n$ monic polynomial, and $A$ the companion matrix. Then there are natural bijections between

• the $A$-invariant subspaces of $K^n$,

• the ideals of $R:=K[X]/(f)$,

• the ideals of $K[X]$ which contain $f$,

• the monic divisors of $f$.

To see this, let $x$ be image of $x$ in $R$, and observe that $A$ is the matrix of the multiplication by $x$ in the basis formed by the degree $< n$ monomials in $x$.

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