# Property of polynomial in $L[x]$ where $L|_k$ is an algebraic extension.

Problem: Let $$L|_k$$ be an algebraic field extension . $$f(x)\in L[x]$$ a polynomial then $$\exists g(x)\in L[x]$$ such that $$f(x)g(x)\in k[x]$$

My approach: It is enough to prove for $$f(x)$$ irreducible in $$L[x]$$. If $$f(x)$$ is irreducible in $$L[x]$$ there is an extension $$L(\alpha)$$ over $$L$$ such that minimal polynomial of $$\alpha$$ over $$L$$ is $$f(x)$$.

Again $$\alpha$$ being algebraic over $$k$$ there is $$h(x)\in k[x]\subseteq L[x]$$ such that $$h(\alpha)=0$$ hence $$f(x)$$ divides $$h(x)$$ in $$L[x]$$. Hence it is done.

Now I am not sure if this argument is correct. Help me if I am wrong. And other approaches are also welcome

• This looks good to me! Jul 28, 2020 at 8:46
• Thanks @Stahl. Other approaches are also welcome Jul 28, 2020 at 8:50