Galois Groups of Polynomials

Question

Computing Galois groups of polynomials is a kind of standard thing to do in algebra, so I think this question goes without much further motivation.

It is known that every element of the Galois group is a permutation of the roots of the irreducible factors of some given polynomial. If we fix a labeling of the roots, this gives an injection from the Galois group into the symmetric group of the appropriate order, which is clearly a homomorphism, as both group operations are composition.

Since every group is a subgroup of a symmetric group, we have the natural question:

Is every group the Galois group of some polynomial (over $\mathbb Q$)?

Now it is also not hard to show using cyclotomic polynomials that every Abelian group is the Galois group for some polynomial over $\mathbb Q$, so in the abelian case, the answer is affirmative. What can be said in the remaining cases?

It is known that every FINITE group is the Galois group of some algebraic extension of $\mathbb Q$. — Piquito, Jun 13 '16 at 01:43
@Piquito: No it isn't, unless you know something we don't... — carmichael561, Jun 13 '16 at 02:51

score 4 · Answer 1 · answered Jun 13 '16 at 01:34

4

This is known as the inverse Galois problem (Wikipedia) and it has been extensively studied, with many partial results, though the problem remains unsolved in full generality.

answered Jun 13 '16 at 01:34

Zev Chonoles

129,973

This is really cool! Are there any standard results in this direction that you could provide a reference for? I'm on summer break and would love to read about this further. – A. Thomas Yerger Jun 13 '16 at 01:37
Probably the first thing to take a look at is Hilbert's irreducibility theorem (Wikipedia). This math.SE thread has some more references to read from. – Zev Chonoles Jun 13 '16 at 01:43

Galois Groups of Polynomials

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