roots of minimal polynomial in a simple field extension.

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March 1st 2013, 06:37 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!
March 1st 2013, 07:24 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?

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