How to show $[F(a):F[a^4+a^2+1]$ is finite.

Question

I have an element $a$ in an extension field $F$.

I'm asked if it is true that $a^4+a^2+1$ is algebraic over $F$ if and only if $a$ is algebraic over $F$. I know that if I can show $[F(a):F(a^4+a^2+1)]$ has a finite degree that $a$ is algebraic and I'll be good. How can I show that it is finite?

Answer 1

First, note that since $1 \in F$ is contained in every field, you have $F(a^4+a^2+1) = F(a^4+a^2)$.

In order to show this degree is finite you only need to prove that $a$ is algebraic over $F(a^4+a^2+1)$ right?

But this is clear because $a$ is the root of the polynomial \begin{equation*} X^4 + X^2 - (a^4+a^2) \in F(a^4+a^2)[X] \end{equation*}

Note: If you don't want to supress the $1$, simply consider the polynomial \begin{equation*} X^4 + X^2 - (a^4+a^2+1)+1 \in F(a^4+a^2+1)[X] \end{equation*}