# Thread: roots of minimal polynomial in a simple field extension.

1. ## roots of minimal polynomial in a simple field extension.

Hi,

Let $\displaystyle K$ be a field and let $\displaystyle L=K[l]$ be a simple field extension of $\displaystyle K$. Let $\displaystyle m_{K,l}$ be the minimal polynomial of $\displaystyle l$ over $\displaystyle K$.

My question is how many roots does $\displaystyle m_{K,l}$ have in $\displaystyle L$?

Clearly, by definition we have that $\displaystyle m_{K,l}(l)=0$. So we at least have one roots (namely $\displaystyle l$ ) but i've got a feeling this is actually the only roots. Is that correct?

Thanks for any help!

2. ## Re: roots of minimal polynomial in a simple field extension.

Originally Posted by Ant
Hi,

Let $\displaystyle K$ be a field and let $\displaystyle L=K[l]$ be a simple field extension of $\displaystyle K$. Let $\displaystyle m_{K,l}$ be the minimal polynomial of $\displaystyle l$ over $\displaystyle K$.

My question is how many roots does $\displaystyle m_{K,l}$ have in $\displaystyle L$?

Clearly, by definition we have that $\displaystyle m_{K,l}(l)=0$. So we at least have one roots (namely $\displaystyle l$ ) but i've got a feeling this is actually the only roots. Is that correct?

Thanks for any help!
You mean "this is not actually the only root", right?