Let $K$ be an extension field of $F$ and let $a$ be an element of $K$. Show that if $F(a)=F(a^2)$, then $a$ is algebraic over $F$.

Question

Question: Let $K$ be an extension field of $F$ and let $a$ be an element of $K$. Show that if $F(a)=F(a^2)$, then $a$ is algebraic over $F$.

Thoughts: I thought this would be simple, but I am having a bit of trouble. Now, we know $[F(a):F(a^2)]=1$, so $[K:F]=[K:F(a^2)][F(a^2):F(a)][F(a):F]=[K:F(a^2)][F(a):F]$, but now I'm stuck. I am not given that $K$ is a finite extension, because then $[F(a):F]$ divides a finite number $[K:F]$ and we're done, but I'm not sure where to go from here. I suppose my next step is to use some field of fraction stuff... any help is greatly appreciated! Thank you.

Answer 1

Unfortunately, your attempted calculations will be difficult to make work, since they all implicitly rely on $[F(a):F]$ being finite.

A hint towards a different approach: If $F(a^2)=F(a)$, then $a\in F(a^2)$; in particular, $a$ can be written as a rational function of $a^2$ with coefficients in $F$. Can you use this to show that $a$ is algebraic?