What (permutation) groups can occur as galois groups of irreducible polynomials of degree n - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:01:12Z http://mathoverflow.net/feeds/question/46679 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46679/what-permutation-groups-can-occur-as-galois-groups-of-irreducible-polynomials-o What (permutation) groups can occur as galois groups of irreducible polynomials of degree n Timothy Wagner 2010-11-19T21:03:14Z 2010-11-20T22:45:48Z <p>I think the answers for the first few degrees ($n$) are:</p> <p>$n=2$, $S_2$</p> <p>$n=3$, $S_3,A_3$</p> <p>$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)</p> <p>$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ($Fr_5$ is a Frobenius group)</p> <p>What results do we have for higher orders and are there any results for a general $n$?</p> <p>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.</p> http://mathoverflow.net/questions/46679/what-permutation-groups-can-occur-as-galois-groups-of-irreducible-polynomials-o/46693#46693 Answer by Joël for What (permutation) groups can occur as galois groups of irreducible polynomials of degree n Joël 2010-11-19T23:43:51Z 2010-11-19T23:43:51Z <p>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 $G$.</p> <p>Conclusion: the subgroups of $S_n$ that are the Galois group of an irreducible polynomial are the transitive ones. </p> http://mathoverflow.net/questions/46679/what-permutation-groups-can-occur-as-galois-groups-of-irreducible-polynomials-o/46706#46706 Answer by Alex Bartel for What (permutation) groups can occur as galois groups of irreducible polynomials of degree n Alex Bartel 2010-11-20T03:09:03Z 2010-11-20T03:09:03Z <p>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:</p> <ul> <li>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.</li> <li>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).</li> <li>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.</li> </ul> <p>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 <a href="http://en.wikipedia.org/wiki/Embedding_problem" rel="nofollow">Embedding problem</a>.</p>