Element algebraic over field iff algebraic over field extension

Question

Let $F$ be a field, and $K$ a finite extension of $F$. I want to show that any element algebraic over $K$ is algebraic over $F$, and conversely.

Well, if $a$ is algebraic over $F$, we can write $c_0+c_1a+\ldots+c_na^n=0$ for some $c_0,\ldots,c_n\in F$. But $c_0,\ldots,c_n\in K$, so $a$ is algebraic over $K$.

Now suppose $a$ is algebraic over $K$, so we can write $c_0+c_1a+\ldots+c_na^n=0$ for some $c_0,\ldots,c_n\in K$. What to do then?

Answer 1

Hint: $$F(a) \subset K(a)$$

And $[K(a):F]=[K(a):K] [K:F] < \infty$. From here you should be able to get that

$$[F(a):F]< \infty \,.$$