In what field do factors of polynomial belong?

Question

In my notes I found the following. Let $f\in F[X]$ and let $u$ be a root of $f$ then $f=(X-u)m$ where $m\in F(u)[X]$. What I don't understand is why $m\in F(u)[X]$?

Thanks

Answer 1

To apply the Euclidean algorithm to divide $P$ by $Q=X-u$, you need for the coefficients of $Q$ and $P$ to lie in the same ground field. If they don't, you can still do it over an extension containing the coefficients, which is precisely what is done here.

Answer 2

With the implicit assumptions added in bold:

Let $F$ be a field. Let $f \in F[X]$ and let $u$ be a root of $f$ in an extension $G$ of $F$ then $f = (X-u) m$ where $m \in F(u)[X]$.

Then it follows from factor theorem.

Answer 3

As I see it I think that $F(u)[X]$ means the ring of polynomials with coefficients from the field $F(u)$

For example say $x^2+1\in\mathbb{R}[X]$ and $u=i$ a root of $f$ , then $f=(x-i)(x+i)$ where $m=x+i\in \mathbb{R}(i)[X](=\mathbb{C}[X])$