Talk:Abel–Ruffini theorem
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[edit] Too much emphasis
Good article but there is a lot of emphasis used. I'd suggest that the author remove all but the most important italics. —Preceding unsigned comment added by Jormundgard (talk • contribs) 21:27, 29 December 2005 (UTC)
[edit] Is this article accurate?
I think this article contains a major historical inaccuracy. It says that Ruffini (and, independently, Abel) proved that the solution of the general polynomial equation of degree ≥ 5 in radicals is impossible. I think what these two showed is that the solution in radicals of the general polynomial equation of degree = 5 is impossible. Evariste Galois is generally recognized as the guy who extended the result to degrees > 5. Isn't that right? DavidCBryant 18:18, 10 January 2007 (UTC)
- If you have a proof for degree 5, the result for degree >5 follows immediately. Specifically, if there were a solution in radicals for ax6 + bx5 + cx4 + dx3 + ex2 + fx + g = 0, then you could just put g=0 and you immediately have a solution in radicals for degree-5 polynomials, which contradicts the Abel-Ruffini theorem. -- Dominus 18:44, 10 January 2007 (UTC)
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- Yeah, I was definitely asleep at the switch when I wrote my first post. Thanks for clearing that up. But I still think something's not quite right, in between this article, the biography of Galois, and the biographies of Ruffini and Abel. I was reading all that stuff yesterday, and I got the distinct impression that the (admittedly confusing) history of the "quintic" problem is not described clearly on Wikipedia. I'll try reading it all over again so I can explain what bugged me a little more precisely. (Oh ... I think this is part of it. Abel-Ruffini establishes the impossibility of a general solution, but does not completely characterize the special cases, such as x8 - 2x4 + 16 = 0, where a solution in radicals is possible. Didn't it take Galois field theory to complete that characterization? I'm not real big on algebra.) DavidCBryant 12:25, 11 January 2007 (UTC)
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- I too take issue with the historical accuracy. Far as I've gathered, Galois only found a method to answear whether a given quintic equation could be solved by radicals, not if it was solvable by other means should the radicals fail. The general result, using what had then become known as "galois theory", came ca 60 years after the proof by Niels Henrik Abel. But I'm not sure who got it first of Abel or Ruffini, though there was a noticeable delay between Abel finishing his proof and having it printed, posthumously. EverGreg 20:35, 3 August 2007 (UTC)
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[edit] Proof Correct
There was a question in the article itself as to whether the proof was correct. It noted that the proof asserted that [E:F] is less than or equal to 5!, whereas what is needed is that |G(E/F)| is less than or equal to 5!. But G(E/F) is the Galois group of the extension E/F, so these are equivalent statements. Thus, the proof is correct as written, and I edited out your concern. --LamilLerran 18:46, 14 February 2007 (UTC)
[edit] Which field
I was reading this page and I think it would be helpful, to myself and other readers, if the author included: 1.) In the proof section, that x is an element of the reals (it is not explicitly mentioned), 2.) Which field F is specifically.- Nmech (talk) 18:44, 22 February 2008 (UTC)
