Talk:Galois theory

message: I have a minor beef with the "Example of a quadratic equation" section.  Roots should be thought of algebraicly and the discussion of symmetries in a graph is misleading.  For instance, we can permute the three roots of x^3-2 but it doesn't correspond to any reflections or rotations of a graph.  Maybe this should be rewritten? dunkstr 01:13, 25 May 2004 (UTC)
replies: text: done --Dmharvey 14:04, 27 May 2005 (UTC)
replies: text: The quadratic example  (x - 2)(x - 1) is not clear, at least to me.  The commentary says that there is only one permutation; to me it looks like there are two,  (1,2)-> (1,2)  and (1,2) -> (2, 1).  Am I missing something?    2001:558:6011:1:38F3:171E:261:EBEA (talk) 19:56, 29 June 2017 (UTC)
replies: text: One of the roots satisfies the equation x-1=0,  which does not remain true if the root are exchanged. Thus the permutation that exchanges the roots does not belong to the Galois group, which, here, is reduced to the identity permutation. D.Lazard (talk) 20:14, 29 June 2017 (UTC)
thread_title: Quadratic example

message: The definition of solvable given in the article is wrong. Cyclic should be replaced with abelian.
159.115.238.165 (talk) 00:15, 8 December 2009 (UTC)
replies: text: There's nothing wrong with it. You are presumably referring to the definition of solvable group as one having a subnormal series with abelian factors. Now, (for finite groups) it does not make a difference whether you use "abelian" or "cyclic" in this definition, as every finite abelian group is a direct sum of cyclic groups. Note that the (equivalent) definition in this article is still different, it asks for a composition series with cyclic factors, not just subnormal series; since factors of a composition series are simple by definition, it cannot have noncyclic abelian factors, thus it would be pointless to refer to general abelian groups here. Moreover, in the context of Galois theory it makes much more sense to refer to cyclic instead of abelian even for subnormal series, as radical field extensions correspond to cyclic factors. — Emil J. 12:41, 8 December 2009 (UTC)
thread_title: Solvable

message: But only in the roots, or am I missing something?  —Preceding unsigned comment added by 87.4.176.2 (talk) 21:39, 1 January 2010 (UTC)
replies: text: You're not missing anything, it was indeed wrong. — Emil J. 14:06, 4 January 2010 (UTC)
thread_title: Discriminant is not a symmetric function in the coefficients

message: thanks for writing this great article
the last sentence of the first paragraph in the section
"solvable groups and solution by radicals"
is very very long and includes two "if" 
It's much too confusing for me!
can you please rewrite it.  —Preceding unsigned comment added by 84.229.239.105 (talk) 20:28, 19 March 2011 (UTC)
thread_title:  Solvable groups and solution by radicals not clear 

message: Under the heading "Inverse Galois problem", the vague phrase "For that" is used. The word "that" might refer to the easy construction, the ground field or something else.  — Preceding unsigned comment added by 31.53.53.179 (talk) 06:25, 6 September 2018 (UTC)
replies: text: . This led me to further edit the section. D.Lazard (talk) 07:29, 6 September 2018 (UTC)
thread_title:  Vague phrase 

message: Pre-history says coefficients are symmetric functions of the roots "up to sign"; looks to me they are functions of the roots either "up to a common factor", or, assuming the highest-order coefficient is fixed to 1, uniquely defined by the roots. Am I missing something?
12.104.156.31 (talk) 00:28, 22 October 2019 (UTC)
replies: text: The assertion in the article is "the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots". This is cocorrect, as "monic" means that the highest-degree coefficient is 1 (you can follow the link). D.Lazard (talk) 08:15, 22 October 2019 (UTC)
thread_title:  up to sign? 

message: In his Oeuvres mathématiques d’Évarist Galois in 1846, Joseph Liouville writes, "... I engaged myself, under the watchful eyes of his brother [Alfred Galois], so to speak, to the attentive study of all the publications and manuscripts he left." So, what motivation did Joseph Liouville have to embark on a careful study of Évarist's papers 14 years after his death? What role did Alfred play in motivating Joseph Liouville to invest his time and effort? OneSkyWalker (talk) 20:20, 14 May 2023 (UTC)
thread_title:  What role did Alfred Galois play in preserving Évariste's legacy?