Direct sum of polynomials in a vector space

Question

I have some trouble with the following problem:

Let be a $E_n$ be the vector space of polynomials in x with coefficients in $\mathbb{C}$ and degree strictly less than n, P and Q two polynomials of degree p and q, p $\ge$ 1 and q $\ge$ 1 without common root.

Let $F_q$ be the set of polynomials of the form AP, A $\in$ $E_q$ and $F_p$ the set of polynomials of the form BQ, Q $\in$ $E_p$.

Show that $E_{p + q}$ is the direct sum of $F_p$ and $F_q$. Infer that there exists a unique couple of polynomials U $\in$ $E_q$ and V $\in$ $E_p$ such that PU + QV = 1.

Thanks a lot for any hint.

Answer 1

Hint 1

Since $P, Q$ are complex polynomials with no common root, they are coprime.

Hint 2

If $A P = B Q$, with $A \in E_{q}$ and $B \in E_{p}$, it follows that $P$ divides $B$, and since $B$ has degree lower than that of $P$, we get $B = 0$, and similarly $A = 0$.

Hint 3

It follows that the sum of $F_{q} = \{ A P : A \in E_{q} \}$ and $F_{p} = \{ B Q : B \in E_{p} \}$ is direct.

Hint 4

The dimension of $F_{q} + F_{p}$ is thus $q + p$, and so is the dimension of $E_{p+q}$.

Hint 5

Since $F_{q} + F_{p} \subseteq E_{p+q}$, they must be equal.

Hint 6

The final statement follows easily.