Galois extension of $\mathbf{Q}(\sqrt[3]{2})$ with specified Galois group

Question

I have a question I've racked my brain over for hours. I want to find a Galois extension of $\mathbf{Q}(\sqrt[3]{2})$ with Galois group $\mathbf{Z}/4\mathbf{Z} \times \mathbf{Z}/8\mathbf{Z}$. More generally what are some ways we can find a Galois extension with a specific Galois group and what conditions on the specified Galois group make such a problem tractible?

Answer 1

If you can find an extension $K/\Bbb Q$ with Galois group $\Bbb Z/4\Bbb Z\times\Bbb Z/8\Bbb Z$ then then $K(\sqrt[3]2)$ will have the same Galois group over $\Bbb Q(\sqrt[3]2)$. To find extensions of $\Bbb Q$ with a given Abelian Galois group, it's useful to take subextensions of cyclotomic fields.

Answer 2

Your last question: "What are some ways we can find a Galois extension [of number fields] with a specific Galois group and what conditions on the specified Galois group make such a problem tractible?" is a very hard problem, known as the inverse Galois problem, as yet unsolved in general, although certain particular cases are known; see e.g. the Wiki article on the subject. I don't know about your background, but the subject surely requires a certain amount of basic knowledge in Galois theory/number theory.

1) Fix a number field $F$. The inverse Galois problem consists, for a group $G$, to find if there exists a Galois extension $L/F$ with group $G$, and possibly to construct it. For various approaches and results, take a look at Wiki on the subject. When $G$ is abelian, a complete answer is given by Class Field Theory. Outside the abelian case, let me cite only the famous theorem of Shafarevich: any solvable group can be realized as a Galois group $Gal(L/\mathbf Q)$. For the "construction" of such a Galois extension, see below.

2) A generalized/related problem is the so called embedding problem. Fix a Galois extension of number fields $K/F$ with group $H$. Given a group $E$ that surjects onto $H$, which we write as an exact sequence $(*)$ $1 \to A \to E \to H \to 1 $ , find a tower of Galois extensions $M/K/F$ with $Gal(M/K)\cong A$, and $Gal(M/F)\cong E$, and the exact sequence of Galois groups is equivalent (in an obvious sense) to the sequence $(*)$. If the kernel $A$ is abelian, $(*)$ is described by a cohomology class in $H ^2(G,A)$ and there are efficient cohomological techniques which allow to bring the problem back to local-global principles (the local problem is much easier).

3) Thus, in principle, the embedding problem with abelian kernel provides the techniques to tackle the « constructive » part of Shafarevich’s theorem. It remains « only » to deal as precisely as possible with the original problem in 1) when $G$ is abelian. For any field $F$ of chacteristic not dividing $n$ and containing the $n$-th roots of unity, where $n$ is the exponent of $G$, Kummer’s theory does the job . For an arbirary number field $F$, it is done by CFT, which classifies and describes the abelian extensions of $F$ using parameters belonging to $F$ alone. For $F=\mathbf Q$, CFT reduces to the Kronecker-Weber theorem which asserts that any abelian number field is contained in a cyclotomic field – hence the hint of @Lord Shark the Unknown.

But since your original question about the group $C_4 \times C_8$ does not involve arithmetic (such as ramification/decomposition of primes), the solution is much simpler and requires only Dirichlet’s theorem on primes in arithmetic progressions. To construct explicitly a polynomial whose Galois group over $\mathbf Q$ is the cyclic group of order $n$, choose a prime $p$ such that $p$ ≡ 1 (mod $n$) and consider the cyclotomic field $\mathbf Q(\zeta)$, where $\zeta$ is a primitive $p$-th root of unity; its Galois group over $\mathbf Q$ is cyclic of order $p − 1$, hence, by the choice of $p$, it contains an (unique) subextension $L$ which is cyclic of degree $n$ over $\mathbf Q$. By taking appropriate sums of conjugates of $\zeta$, following the construction of Gaussian periods, one can find an element of $L$ that generates $L$ over $\mathbf Q$, and compute its minimal polynomial. For more details, see Wiki, op. cit.