$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Question

Let $F$ be a field and $K$ be an extension field of $F$.

Proof\Counterexample: $K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

I haven't been able to determine if it is true or not. My thoughts so far:

$[F(ab):F]\lt \infty$ (is finite)

There must exist some polynomial $p(x)\in F[x]$ such that $p(ab)=0$.

We have to go in one of two directions, right? Either we show that $[F(a)(b):F(a)]\lt \infty$ (is finite) or find some polynomial $g(x)\in F(a)[x]$ so that $g(b)=0$. I'm having trouble showing that it is finite or finding a counterexample if not. Any help would be great.

Answer 1

Let $p(x) \in F[x]$ be such that $p(ab) = 0$. Let $$p(x) = \sum_{i=0}^{n}\alpha_{i}x^{i}$$ where $\alpha_{i} \in F$. Now consider $$g(x) = \sum_{i=0}^{n}\alpha_{i}a^{i}x^{i} \in F(a)[x]$$

What is $g(b)$?