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 $[K(\alpha^2):K(\alpha)]$?

Let $f(X)\in K[X]$ (resp. $g(X$)) be the minimal polynomial of $\alpha$ (resp. $\alpha^2$) over $K$. Because of the degree formula in a tower, $[K(\alpha^2):K] =1,2$ or $4$. The value $1$ is excluded because it would mean that $\alpha^2\in K$, hence $[K(\alpha):K] =2$, contrary to the hypothesis. 
Suppose $[K(\alpha^2):K]=4$, so that $K(\alpha)=K(\alpha^2)$. This would imply that the $K$-homomorphism $K(\alpha)\to K(\alpha^2)$ determined by $\alpha \to \alpha^2$ is an automorphism, in other words, the homomorphism $K[X]\to K[X]$ determined by $X\to X^2$ transforms the principal ideal $(f(X))$ onto $(g(X))$: contradiction with the hypothesis that $\alpha^2$ is not a root of $f(X)$. In conclusion, $[K(\alpha^2):K]=2$.
