# Is the minimal polynomial also minimal over the closure of the base field

field-theorylinear algebraminimal-polynomials

Let $$V$$ be a finite dimensional vector space over the field $$F$$ and $$T: V\to V$$ be linear. Then the minimal polynomial of $$T$$ is the least degree monic polynomial in $$F[x]$$ that annihilates $$T$$. Is it true that it must also be the least degree monic annihilator of $$T$$ in $$\overline F[x]$$?

I know this is true when $$F=\mathbb{R},\overline F =\mathbb{C}$$, but only by seeing $$\mathbb{C}$$ as a finite dimensional vector space over $$\mathbb{R}$$, but I wouldn't know how to extend that to arbitrary fields.
I also tried applying the division algorithm on the minimal polynomial $$m_A \in F[x]$$ and a hypothetical minimal polynomial $$\mu_A \in \overline F[x]$$ of lesser degree to reach a contradiction.

#### Best Answer

Yes, it is true. Take a standard basis $$e_1,...,e_n$$ and let $$e=e_i$$. Consider the sequence of vectors $$e, Te, T^2e,...,T^m e,...$$, all these vectors have coordinates in $$F$$.

There exists the minimal $$m=m(e)$$ such that the vector $$T^me$$ is a linear combination $$\lambda_0e+\lambda_2Te+...+\lambda_{m-1}T^{m-1}e$$ (over $$\bar F$$). Note that since $$T^k e$$ have coordinates in $$F$$, all $$\lambda_i$$ are in $$F$$. Let $$p_e$$ be the polynomial $$t^m-\lambda_{m-1}t^{m-1}-...-\lambda_0$$. Note that it is in $$F[t]$$. Then $$p_e(T)e=0$$.

If $$e_1,...,e_n$$ is the standard basis then the minimal polynomial $$p(t)$$ of the matrix $$T$$ is the lcm of all $$p_{e_i}(t)$$, and so it belongs to $$F[t]$$ too.