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!

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

Quote:

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?