Calculate $[K(\alpha^2):K]$ Let $K\subseteq F$ be a field extension. $K(\alpha)/\ K$ an extension of degree $4$  where $\alpha^2$ is not a root of $m_{(\alpha,K)}(x)$. I am asked to calculate $[K(\alpha^2):K]$
Using the towers formula $$[K(\alpha^2):K]=[K(\alpha^2):K(\alpha)][K(\alpha):K]$$
And we already know that $[K(\alpha):K]=4$
How do I find $[K(\alpha^2):K(\alpha)]$?

The correct tower formula is
$$4=[K(\alpha):K]=[K(\alpha):K(\alpha^2)][K(\alpha^2):K]$$
Since $[K(\alpha):K(\alpha^2)]\le2$, we have $[K(\alpha^2):K]\ge 2$ and so there are two possibilities:


*

*$[K(\alpha^2):K]=2$: this happens iff $K(\alpha^2)\ne K(\alpha)$. For instance, for $K=\mathbb Q$ and $\alpha$ a root of $x^4-2$. Then $\alpha^2$ is a root of $x^2-2$ and is not a root of the minimal polynomial of $\alpha$.

*$[K(\alpha^2):K]=4$: this happens iff $K(\alpha^2)=K(\alpha)$. For instance, for $K=\mathbb Q$ and $\alpha$ a root of $x^4-2x-2$. Then $\alpha^2$ is a root of $x^4 - 4 x^2 - 4 x + 4$, which is irreducible, and so $\alpha^2$ is not a root of the minimal polynomial of $\alpha$.
$x^4-2x-2$ was chosen because then clearly $\alpha$ is a polynomial in $\alpha^2$ and so $K(\alpha^2)=K(\alpha)$. It is irreducible by Eisenstein's criterion with $p=2$.
$x^4 - 4 x^2 - 4 x + 4$ is the characteristic polynomial of the map $x \mapsto \alpha^2 x$ and so must be irreducible since $\alpha^2$ has degree $4$.
