# Thread: Long division and extension fields

1. ## Long division and extension fields

I need some help understanding the following worked example:

So they have $\displaystyle \alpha = x +I$ where I = <p(x)>. But what do they mean "by long division we have..."? Did they divide p(x) by $\displaystyle \alpha$? If so how could they end up with $\displaystyle (x-\alpha) (x+2+\alpha)$? Any help is really appreciated.

2. Originally Posted by demode
I need some help understanding the following worked example:

So they have $\displaystyle \alpha = x +I$ where I = <p(x)>. But what do they mean "by long division we have..."? Did they divide p(x) by $\displaystyle \alpha$? If so how could they end up with $\displaystyle (x-\alpha) (x+2+\alpha)$? Any help is really appreciated.

You know (or should to) that $\displaystyle \alpha:= x+I$ is a root of $\displaystyle p(x)\Longrightarrow (x-\alpha)\mid p(x)$ , and thus you can

(long) divide these two polynomials.

Note that the remainder of this division is $\displaystyle \alpha^2+2\alpha+3=0$ , as expected.

Tonio