Vtyjkyuk

For $E=mathbbQ(i,2^1/8)$ and $F=mathbbQ(i)$, show that $mathrmGal(E/F)$ is cyclic.

I know is that $mathrmGal(E/F)$ is of order 8, since $x^8-2$ is the minimal polynomial of $2^1/8$. I can see that the roots of the minimal polynomial are $2^1/8$,$2^1/8xi$, $2^1/8xi^2$, $2^1/8xi^3$, $2^1/8xi^4...$ where $xi$ is the 8th root of unity. From there, how can I proceed?

Thanks

First, choose a distinguished primitive $8$th root of unity by setting $xi = fracsqrt22(1+i)$. (This choice won't affect our computation of $G := mathrmGal(E/F)$, but rather how we label the roots of $X^8-2$.)

We can be very explicit in computing the $G$ by using the following facts/observations:

(1) $E = F(sqrt[8]2)$; hence, any field automorphism of $E$ which fixes $F$ is determined by where it sends $sqrt[8]2$.

(2) $G$ acts transitively on the roots of $X^8-2$; hence, for each $i = 0, 1, ldots, 7$, there is an element $sigma_i in G$ which sends $sqrt[8]2$ to $xi^isqrt[8]2$. Moreover, each $sigma_i$ is a distinct element of $G$, since $sigma_i(sqrt[8]2) neq sigma_j(sqrt[8]2)$ for distinct $i, j in 0, 1, ldots, 7$.

(3) $|G| = [E:F] = 8$, so in fact, $G = sigma_0, sigma_1, ldots, sigma_7$ by observation (2).

Hence, we have computed the elements of $G$. Now we need to show that $G$ is cyclic. Indeed, it suffices to show that $G$ contains an element of order $8$. I claim that $sigma_1$ has order $8$. The key thing to do is to determine what $sigma_1$ does to $xi$, so let's do that and see how it helps.

Since $sigma_1$ fixes $F$, all we need to do is check what $sigma_1(sqrt2)$ is. This is easy enough: we have

$$sigma_1(sqrt2) = sigma_1((sqrt[8]2)^4) = sigma_1(sqrt[8]2)^4 = (xi sqrt[8]2)^4 = xi^4sqrt2 = -sqrt2.$$

Hence, $sigma_1(xi) = -xi$. You can then check that

$$sigma_1^2(sqrt[8]2) = sigma_1(xisqrt[8]2) = sigma_1(xi)sigma_1(sqrt[8]2) = -xi^2sqrt[8]2$$

I leave the computation that $sigma_1^4(sqrt[8]2) = -sqrt[8]2$ to you; it is similar.

By Lagrange, $sigma_1$ has order $1, 2, 4$ or $8$. Since we see that $sigma_1$ cannot have order $1, 2, $ or $4$, it must have order $8$ as desired. (You can explicitly compute the powers of $sigma_1$ if you like, too.)