Product of polynomes in extensions Sorry if this is an elementary question. Let $K$ be a field and $L$ an extension of $K$. Let $P,Q\in{K[X]}$, so that $Q$ divides $P$ in $L[X]$, i.e. so that there exists a polynome $R\in{L[X]}$ so that $P=QR$. Then $R\in{K[X]}$.
I think this is a consequence of the unique factorisation of $P$ as a product of irreducible polynomes of $K[X]$, but I want to be sure. 

The relation $P=QR$ gives a linear system for the coefficients of $R$.
This system has entries in $K$ because $P,Q \in K[x]$.
If this system has a solution in $L$, then this solution can be found using Gaussian elimination. Therefore, the solution is in $K$ because Gaussian elimination only uses rational operations on the entries and these operations never leave $K$, which is a field.
