# roots of minimal polynomial in a simple field extension.

• Mar 1st 2013, 05:37 AM
Ant
roots of minimal polynomial in a simple field extension.
Hi,

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

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

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

Thanks for any help!
• Mar 1st 2013, 06:24 AM
HallsofIvy
Re: roots of minimal polynomial in a simple field extension.
Quote:

Originally Posted by Ant
Hi,

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

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

Clearly, by definition we have that $m_{K,l}(l)=0$. So we at least have one roots (namely $l$ ) 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?