Timeline for What (permutation) groups can occur as galois groups of irreducible polynomials of degree n
Current License: CC BY-SA 2.5
18 events
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| Nov 21, 2010 at 2:05 | comment | added | Alex B. | For some reason, the link gives a connection timeout for me. Anyway, your question is still ambiguous: the title says "irreducible polynomials", the body says Galois groups over extension of Q of polynomials defined over Z. So over what do you want them to be irreducible, Q or the base field? | |
| Nov 21, 2010 at 0:52 | comment | added | Timothy Wagner | @Alex: This was motivated by problem 5 here math.berkeley.edu/~serganov/114/galsol.pdf I may have interpreted something incorrectly though. | |
| Nov 20, 2010 at 23:05 | comment | added | Alex B. | Your edit concerns a very artificial situation. If the base field is an extension of the rationals, then why is $f$ required to be defined over $\mathbb{Z}$ and not over the ring of integers of that extension? | |
| Nov 20, 2010 at 22:45 | history | edited | Timothy Wagner | CC BY-SA 2.5 |
added 234 characters in body; added 91 characters in body; deleted 4 characters in body
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| Nov 20, 2010 at 7:00 | comment | added | Gerry Myerson | @Kevin, yes, I forgot about that - but everyone believes that problem has a positive solution, no? | |
| Nov 20, 2010 at 3:09 | answer | added | Alex B. | timeline score: 5 | |
| Nov 20, 2010 at 2:38 | comment | added | Andrés E. Caicedo | In case anybody is curious: "All Infinite Groups are Galois Groups Over any Field", Manfred Dugas and Rüdiger Göbel, Transactions of the American Mathematical Society, Vol. 304, No. 1, (Nov., 1987), pp. 355-384. | |
| Nov 19, 2010 at 23:51 | comment | added | Andrés E. Caicedo | If the question is: For which $G$ there is some $K$ and a Galois extension $F/K$ with Galois group $G$, then the answer is any $G$ (even if $G$ is infinite!). Is this what you are asking? | |
| Nov 19, 2010 at 23:43 | answer | added | Joël | timeline score: 14 | |
| Nov 19, 2010 at 22:35 | comment | added | Qiaochu Yuan | @Timothy: are you asking about a particular base field or over an arbitrary base field? | |
| Nov 19, 2010 at 22:23 | comment | added | Kevin Buzzard | Gerry---if you can prove that you've solved the inverse Galois problem! | |
| Nov 19, 2010 at 22:05 | answer | added | user631 | timeline score: 9 | |
| Nov 19, 2010 at 21:59 | comment | added | Gerry Myerson | It's "if and only if," isn't it? $G$ is the Galois group of an irreducible polynomial of degree $n$ (over, say, the rationals) if and only if it's a transitive subgroup of $S_n$. | |
| Nov 19, 2010 at 21:27 | history | edited | Andrey Rekalo | CC BY-SA 2.5 |
Spelling corrected.
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| Nov 19, 2010 at 21:25 | comment | added | Timothy Wagner | Yes, because it acts transitively on the roots. $K_4$ is the Klein group/Klein-four group/Vierergruppe. | |
| Nov 19, 2010 at 21:23 | history | edited | Timothy Wagner | CC BY-SA 2.5 |
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| Nov 19, 2010 at 21:18 | comment | added | Joël | Surely, the group has to be transitive (that is, as a subgroup of $S_n$, has to act transitively on {1,..,n}). By the way, what is K4 in your list? | |
| Nov 19, 2010 at 21:03 | history | asked | Timothy Wagner | CC BY-SA 2.5 |