# Thread: Characteristic polynomial of left-multiplication transformation

1. ## Characteristic polynomial of left-multiplication transformation

Let $\displaystyle K$ be a field, $\displaystyle [K:\mathbb{Q}]=n$, and take $\displaystyle \alpha \in K$ such that $\displaystyle \alpha$ has degree $\displaystyle d$ over $\displaystyle \mathbb{Q}$. Let $\displaystyle f(x)$ be the monic irreducible polynomial over $\displaystyle \mathbb{Q}$ having $\displaystyle \alpha$ as a root.

Let $\displaystyle T_\alpha : K \rightarrow K$ be given by $\displaystyle k \mapsto \alpha k$. Show that the characteristic polynomial of $\displaystyle T_\alpha$ is $\displaystyle f(x)^{n/d}$.

2. Originally Posted by Bruno J.
Let $\displaystyle K$ be a field, $\displaystyle [K:\mathbb{Q}]=n$, and take $\displaystyle \alpha \in K$ such that $\displaystyle \alpha$ has degree $\displaystyle d$ over $\displaystyle \mathbb{Q}$. Let $\displaystyle f(x)$ be the monic irreducible polynomial over $\displaystyle \mathbb{Q}$ having $\displaystyle \alpha$ as a root.

Let $\displaystyle T_\alpha : K \rightarrow K$ be given by $\displaystyle k \mapsto \alpha k$. Show that the characteristic polynomial of $\displaystyle T_\alpha$ is $\displaystyle f(x)^{n/d}$.
from the definition of $\displaystyle T_{\alpha},$ it's easily seen that for any $\displaystyle g(x) \in \mathbb{Q}[x]$ and $\displaystyle k \in K$ we have $\displaystyle g(T_{\alpha})k=g(\alpha)k.$ thus $\displaystyle g(T_{\alpha})=0$ if and only if $\displaystyle g(\alpha)=0.$ in particular $\displaystyle T_{\alpha}$ and $\displaystyle \alpha$ have the same minimal

polynomials. so $\displaystyle f(x)$ is the minimal polynomial of $\displaystyle T_{\alpha}.$ now by, Cayley-Hamilton, the irreducible factors of the characteritsic polynomial and the minimal polynomial of a linear transformation are

the same. thus the caracteristic polynomial of $\displaystyle T_{\alpha}$ is $\displaystyle (f(x))^r,$ for some integer $\displaystyle r \geq 1.$ comparing the degrees gives us $\displaystyle r=\frac{n}{d}.$