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 \$\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!
• Mar 1st 2013, 06:24 AM
HallsofIvy
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?