I know how to do this in the case of $\mathbb{Z}/3\mathbb{Z}$, by looking at subfields of cyclotomic extensions. Specifically, consider the extension $\mathbb{Q}(\zeta_7)$ where $\zeta_7$ is a primitive $7^{th}$ root of unity. The Galois group is the cyclic group on 6 elements. We know that $\zeta_7+\zeta_7^{-1}$ is fixed by complex conjugation, and since complex conjugation generates a subgroup of index 3, we may simply take the fixed field of this subgroup. Note that since the Galois group is cyclic, it is abelian, thus every subgroup is normal so there are no issues here. It also turns out that complex conjugation is the only automorphism fixing $\zeta_7+\zeta_7^{-1}$, so our fixed field is precisely $\mathbb{Q}(\zeta_7+\zeta_7^{-1})$. My question is, can we find an analogue for $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$?
Find a Galois extension $\mathbb{Q}\subset K$ with $\text{Gal}(K/\mathbb{Q})\cong\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$
abstract-algebrafield-theorygalois-theory
Related Solutions
Somehow, the theme of symmetrization often doesn't come across very clearly in many expositions of Galois theory. Here is a basic definition:
Definition. Let $F$ be a field, and let $G$ be a finite group of automorphisms of $F$. The symmetrization function $\phi_G\colon F\to F$ associated to $G$ is defined by the formula $$ \phi_G(x) \;=\; \sum_{g\in G} g(x). $$
Example. Let $\mathbb{C}$ be the field of complex numbers, and let $G\leq \mathrm{Aut}(\mathbb{C})$ be the group $\{\mathrm{id},c\}$, where $\mathrm{id}$ is the identity automorphism, and $c$ is complex conjugation. Then $\phi_G\colon\mathbb{C}\to\mathbb{C}$ is defined by the formula $$ \phi_G(z) \;=\; \mathrm{id}(z) + c(z) \;=\; z+\overline{z} \;=\; 2\,\mathrm{Re}(z). $$ Note that the image of $\phi$ is the field of real numbers, which is precisely the fixed field of $G$. This example generalizes:
Theorem. Let $F$ be a field, let $G$ be a finite group of automorphisms of $F$, and let $\phi_G\colon F\to F$ be the associated symmetrization function. Then the image of $\phi_G$ is contained in the fixed field $F^G$. Moreover, if $F$ has characteristic zero, then $\mathrm{im}(\phi_G) = F^G$.
Of course, since $\phi_G$ isn't a homomorphism, it's not always obvious how to compute a nice set of generators for its image. However, in small examples the goal is usually just to produce a few elements of $F^G$, and then prove that they generate.
Let's apply symmetrization to the present example. You are interested in the field $\mathbb{Q}(\zeta_7)$, whose Galois group is cyclic of order $6$. There are two subgroups of the Galois group to consider:
The subgroup of order two: This is the group $\{\mathrm{id},c\}$, where $c$ is complex conjugation. You have already used your intuition to guess that $\mathbb{Q}(\zeta_7+\zeta_7^{-1})$ is the corresponding fixed field. The basic reason that this works is that $\zeta_7+\zeta_7^{-1}$ is the symmetrization of $\zeta_7$ with respect to this group.
The subgroup of order three: This is the group $\{\mathrm{id},\alpha,\alpha^2\}$, where $\alpha\colon\mathbb{Q}(\zeta_7)\to\mathbb{Q}(\zeta_7)$ is the automorphism defined by $\alpha(\zeta_7) = \zeta_7^2$. (Note that this indeed has order three, since $\alpha^3(\zeta_7) = \zeta_7^8 = \zeta_7$.) The resulting symmetrization of $\zeta_7$ is $$ \mathrm{id}(\zeta_7) + \alpha(\zeta_7) + \alpha^2(\zeta_7) \;=\; \zeta_7 + \zeta_7^2 + \zeta_7^4. $$ Therefore, the corresponding fixed field is presumably $\mathbb{Q}(\zeta_7 + \zeta_7^2 + \zeta_7^4)$.
All that remains is to find the minimal polynomials of $\zeta_7+\zeta_7^{-1}$ and $\zeta_7 + \zeta_7^2 + \zeta_7^4$. This is just a matter of computing powers until we find some that are linearly dependent. Using the basis $\{1,\zeta_7,\zeta_7^2,\zeta_7^3,\zeta_7^4,\zeta_7^5\}$, we have $$ \begin{align*} \zeta_7 + \zeta_7^{-1} \;&=\; -1 - \zeta_7^2 - \zeta_7^3 - \zeta_7^4 - \zeta_7^5 \\ (\zeta_7 + \zeta_7^{-1})^2 \;&=\; 2 + \zeta_7^2 + \zeta_7^5 \\ (\zeta_7 + \zeta_7^{-1})^3 \;&=\; -3 - 3\zeta_7^2 - 2\zeta_7^3 - 2\zeta_7^4 - 3\zeta_7^5 \end{align*} $$ In particular, $(\zeta_7+\zeta_7^{-1})^3 + (\zeta_7+\zeta_7^{-1})^2 - 2(\zeta_7+\zeta_7^{-1}) - 1 = 0$, so the minimal polynomial for $\zeta_7+\zeta_7^{-1}$ is $x^3 + x^2 - 2x - 1$. Similarly, we find that $$ (\zeta_7 + \zeta_7^2 + \zeta_7^4)^2 \;=\; -2 - \zeta_7 - \zeta_7^2 - \zeta_7^4 $$ so the minimal polynomial for $\zeta_7 + \zeta_7^2 + \zeta_7^4$ is $x^2+x+2$.
The Galois group of $F/\mathbb Q$ is cyclic of order $6$ as you described above, and if $\sigma$ is a generator for this group, then since the unique subgroup of order $2$ is $\{1, \sigma^2, \sigma^4 \}$, we can assume without loss of generality that $\sigma^2 = u$. If you compute $\sigma$, you can compute $\sigma^2$ and explicit $E$ as the fixed field of $\sigma^2$.
Now the extension $F/\mathbb Q$ is simple, so $\sigma$ is characterized entirely by where it maps $f$ (let me write $f = \zeta_7$ for clarity). It suffices to find a primitive root of $(\mathbb Z/7\mathbb Z)^{\times}$ and we got it. After some testing you realize that $3^6 = 1$ but $3^2, 3^3 \neq 1 \pmod 7$, so $3$ is a primitive root. This means that $\sigma(\zeta_7) = \zeta_7^3$ is a possibility for $\sigma$ as a generator of the Galois group.
Now $\sigma^2(\zeta_7) = \zeta_7^9 = \zeta_7^2$ (as promised by $\sigma^2 = u$ and $u(f) = f^2$), so we're trying to find the fixed field corresponding to the group $\{ 1,\sigma^2,\sigma^4\}$.
Because of the simple relation $\zeta_7^7 = 1$, it is not hard to solve the equation $\sigma^2(x)-x = 0$ for $x \in F$ ; write $$ x = a_0 + a_1 \zeta_7 + \cdots + a_5 \zeta_7^5 $$ and apply $\sigma^2$ to it ; the equation $\sigma^2(x) = x$ will give you linear conditions on the coefficients. I'm afraid finding generators involves linear algebra. If you have the patience of writing down the equation by hand, you'll see the computations do not require any smart-ideas and you get $$ E = \mathbb Q(\zeta_7 + \zeta_7^2 + \zeta_7^4). $$ The minimal polynomial can be computed, it is $$ (X - (\zeta_7 + \zeta_7^2 + \zeta_7^4))(X - (\zeta_7^3 + \zeta_7^5 + \zeta_7^6)) = X^2 + X + 2. $$ The roots of this polynomial are $\frac{-1 \pm \sqrt{-7}}2$, so you can write $$ E = \mathbb Q(\sqrt{-7}). $$ Note : The magical reason why I get only one canonical element (i.e. $\zeta_7 + \zeta_7^2 + \zeta_7^4$) as a generator for $E/\mathbb Q$ is because $[E:\mathbb Q] = 2$, so I know the extension can only be generated by one element and that a basis for $E/\mathbb Q$ will be given by $\{1,x\}$ for some $x \in E$. This means when I tried to compute the fixed field of $\sigma^2$, I know the term for $1$ would be irrelevant and the rest would only give me one term.
Interesting remark : You know that $\mathbb Z / 6 \mathbb Z$ has only one subgroup of index $2$ and that the extension $F/\mathbb Q$ is Galois, so by the Galois correspondence $F$ has only one subfield of order $2$ over $\mathbb Q$. The element $\zeta_7 + \sigma^2(\zeta_7) + \sigma^4(\zeta_7)$ is invariant by $\sigma^2$ and is not in $\mathbb Q$, so you could've skipped the linear algebra computations and know already that $E = \mathbb Q(\zeta_7 + \zeta_7^2 + \zeta_7^4)$. We don't escape the computation of the minimal polynomial to simplify this though. (These ideas of averaging are often used in invariant theory ; I averaged the orbit of $\zeta_7$ over the subgroup $\{1,\sigma^2,\sigma^4\}$ to find the invariant substructure I wanted.)
Hope that helps,
Best Answer
Hint:
Let $\mu_n$ denote the set of $n$-th roots of unity. If $n,m$ are coprime, then $\mathrm{Gal}(\mathbb{Q}(\mu_{nm})/\mathbb{Q})\cong\mathrm{Gal}(\mathbb{Q}(\mu_n)/\mathbb{Q})\times\mathrm{Gal}(\mathbb{Q}(\mu_m)/\mathbb{Q})$ via restrictions. Using this, you can find a Galois extension with Galois group $\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ and then you can conclude via the fundamental theorem of Galois theory. (It may be worthwhile to note that if you know the sturcture theorem for finite abelian groups and a special case of Dirichlet's theorem on arithmetic progressions, then you can prove that any abelian group can be realized as Galois group over $\mathbb{Q}$ using this method).