Results 1 to 2 of 2

Thread: Characteristic polynomial of left-multiplication transformation

  1. #1
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    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 $\displaystyle f(x)^{n/d}$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Bruno J. View Post
    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 $\displaystyle r=\frac{n}{d}.$
    Last edited by NonCommAlg; Jun 16th 2009 at 02:59 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Characteristic Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Nov 28th 2011, 06:01 AM
  2. Characteristic Polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Apr 21st 2010, 02:40 PM
  3. characteristic polynomial
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Apr 6th 2010, 04:01 AM
  4. Replies: 1
    Last Post: Dec 15th 2009, 07:26 AM
  5. Characteristic polynomial
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Nov 13th 2006, 07:35 AM

Search Tags


/mathhelpforum @mathhelpforum