Galois extension, simple extension, power

Question

Hello,

let $k$ be a field with characteristec zero.

and suppose $k$($\alpha$) is Galois extension of $k$.

then, is it true $k$($\alpha^3$) = $k$($\alpha$) ? or not?

Thank you for your help.

Answer 1

Not necessarily. For example, if $k=\mathbb{Q}$ and $\alpha$ is a primitive third root of unity, i.e. a root of $x^2 + x + 1 = 0$, so $\alpha = \frac{-1+\sqrt{-3}}{2}$ and $\alpha^3 = 1$, then $k(\alpha)$ is a quadratic extension of $k$; but $k(\alpha^3)=k(1)=k$.

On the other hand, if $\alpha$ is $i$, i.e. a primitive 4th root of unity, then $\alpha^3 = -i$, so $k(\alpha) = k(i) = k(-i)= k(\alpha^3)$.

Since you're looking at it from a Galois theoretic perspective: as idm commented, $k(\alpha^3)$ must be inside $k(\alpha)$, since $\alpha^3\in k(\alpha)$. However, it is not the whole thing if $\alpha^3$ is fixed by any nontrivial elements of the Galois group of $k(\alpha)$ over $k$. In my first example, $\alpha^3 = 1$ was fixed by the whole Galois group because it is rational. In my second example, $\alpha^3 = -i$ is moved by any group element that moves $\alpha = i$, so no nontrivial element of the Galois group can fix it.