Question

I think the answers for the first few degrees ($n$) are:

$n=2$, $S_2$

$n=3$, $S_3,A_3$

$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)

$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ($Fr_5$ is a Frobenius group)

What results do we have for higher orders and are there any results for a general $n$?

Edit: Sorry, I have aptly demonstrated that I am still rough on some of my concepts here. I believe I was thinking about the case where $f(x)\in \mathbb{Z}[x]$ and the base field is a finite extension of the rational numbers. I hope I have gotten it right this time and the above answers are right for this situation.

Answer 1

Let $G$ be a transitive subgroup of $S_n$. Let $L=\mathbb{C}(X_1,\dots,X_n)$. The group $G$ acts on $L$ by permutations of the variables. Let $K=L^{G}$. Then $L/K$ is Galois of Galois group $G$. Let $P$ be the minimal polynomial of $X_1$ over $G$. It is irreducible. It has degree at least $n$ because all the $X_i$ are Galois conjugate of $X_1$ over $K$, by transitivity of $G$. On the other hand, it divides $(X-X_1)\dots(X-X_n)$ which has coefficients in $L^{S_n} \subset K$. Hence $P$ has degree $n$. The splitting field of $P$ over $K$ is $L$ obviously, so its Galois group is $G$.

Conclusion: the subgroups of $S_n$ that are the Galois group of an irreducible polynomial are the transitive ones.

Answer 2

This question is ambiguous and can be understood in various ways, since no mention is made of the base field. If you are asking "for what subgroups $G$ of $S_n$ do there exist fields $L,K$ such that $L/K$ is a Galois with Galois group $G$", then Joël has given a complete answer. If you fix the base field $K$ and ask what groups can arise as Galois groups over $K$, then the question can be anything between trivial and open, depending on $K$. Here are are some examples of the spectrum of hardness:

• If the base field is finite, then of course finite $G$ can be realised as a Galois group if and only if $G$ is cyclic.
• If $K$ is a local field with finite residue field, then a necessary condition for $G$ to be realisable as a Galois group over $K$ is that $G$ be solvable. There are other restrictions as well, coming from the filtration on $G$ by higher ramification groups. However, not all finite solvable groups can be realised over a given local field. A fun exercise (if you know about the structure of higher ramification groups) is: classify all primes $p$ and $q$ such that the dihedral group $D_{2q}$ is realisable as a Galois group over the field $\mathbb{Q_p}$. Actually, the absolute Galois groups of local fields are fairly well understood. They are certain pro-solvable groups explicitly given by generators and relations (see Neukirch's book on Cohomology of Number fields). So in principle, given any finite group, you could try checking whether it's a quotient of the absolute Galois group (which might still be very difficult).
• If $K$ is a number field, then the question you are posing is a famous open problem. Shafarevich proved that all solvable finite groups are Galois groups over $\mathbb{Q}$ and people are chipping away at special cases of insoluble groups, but so far, no uniform picture is apparent.

Note, that a generalisation of the Inverse Galois Problem over a given field, which should help answering the inverse Galois problem over all finite extensions of your given field, is the so-called Embedding problem.