Prove that $K \otimes_F F(\theta) \cong K[x]/(p(x))$ as $K$-algebras.

Let $F \subseteq K \subseteq L$ and let $\theta \in L$ with $p(x) = m_{\theta, F}(x).$ Prove that $K \otimes_F F(\theta) \cong K[x]/(p(x))$ as $K$-algebras.

I've been trying to work through chapters 13/14 in Dummit & Foote. I came across this problem in section 14.4 and I'm not sure quite what to do since I'm a bit unfamiliar with tensor products. Any ideas to help out?

• Well, $F(\theta) \cong F[x]/(p)$ as $F$ algebras, and the set of scalars are formally exchanged to $K$, to make it (freely) a $K$-algebra. – Berci Mar 28 '18 at 19:23

1 Answer

As it is pointed out in the comment, $F(\theta)\cong F[x]/(p)$. Consider the $K$-algebra map $\varphi : K\otimes_F F[x]/(p)\rightarrow K[x]/(p)$ with $\varphi(c\otimes\overline{q})=c\cdot\overline{q}$. Can you show that it is an isomorphism?