Talk:Galois theory

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 Field: Algebra

Compass and straightedge[edit]

Hi. Great stuff I do not know a lot about. And good to read. I'd like to add one little thought. When you write constructed with a straight edge and a compass shouldn't that be constructed with a compass? What do you think? Cheers Robert_Dober 21:33 Oct 17, 2002 (UTC)

Automorphism group versus Galois group[edit]

This page needs attention at the point where it defines Aut(L/K) to be a Galois group. Well, it is that precisely when it's a Galois extension; the old treatment here at WP seems not really adequate on this matter.

Charles Matthews 10:07, 6 Feb 2004 (UTC)

Quadratic example[edit]

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)

done --Dmharvey 14:04, 27 May 2005 (UTC)

Bold changes[edit]

Dear all

I have made major changes to the articles in the Galois theory category. I am a wikipedia newbie, but I followed the instructions "BE BOLD". Already while I am editing someone else is making simultaneous changes and confusing the hell out of me!! Excellent!!!

Here is a summary of what I've done so far.

For the article Galois theory, Galois theory article

  • rewrote introduction
  • rewrote classical problems section
  • converted "symmetry groups" + "quadratic polynomial example" into more detailed and accurate "first example" (this addresses an issue mentioned by Dunkstr above
  • polished "second example" - by the way I really like this example, it very carefully keeps away from issues that a person with less algebra skills would have; where is it from?
  • rewrote "modern approach by field theory" section, in particular including adding section on "advantages of the modern approach"
  • minor changes to Inverse Problems section

Reorganised the article on Galois extensions.

On Galois groups, reorganised a bit, included link back to Galois theory for "more elementary examples".

Updated the "Galois theory" category page.

--Dmharvey 13:50, 27 May 2005 (UTC)

And now here is a list of further changes I would like to see happen or at least be discussed:

  • would be nice to have more detailed historical information about the passage from pre-Galois ideas (which DID include some permutation groups) to Galois's ideas to the field theoretic approach
  • the page on quintic equations is almost non-NPOV :-) in terms of insisting that algebraic solutions to quintics exist, although I concede the material is accurate (as far as I know). However I'm not sure if the historical information there is accurate, and it should have links to the abel-ruffini theorem.
  • there seems to be a systematic bias in many of the mathematics pages in favour of assuming all polynomials are defined over the real numbers. I'm not sure what to do about this, since that's all the lay audience knows about; on the other hand, surely it is possible to increase accuracy without losing readability for this majority of people.
  • somewhere need to discuss relationship between galois theory and analogous theories, e.g. covering spaces in topology and/or riemann surfaces. Perhaps this belongs with Galois connections; but I feel it can be mentioned directly on the Galois theory page.
  • really need to add references to some non-online materials, standard books on galois theory, my favourite off the top of my head is chapter 4 of jacobson's basic algebra I, and there are millions of others
  • my list of the advantages of the modern approach to galois theory is heavily number-theory biased.
  • I would like an in-depth discussion of infinite galois groups somewhere.
  • There seems to be quite a lot of duplication on the topic of the abel-ruffini theorem. I think perhaps the section on solvable groups in Galois theory should be merged into the Abel-Ruffini page, with appropriate links to the Solvable groups page.
  • Need to merge inverse problems section on Galois theory page with the single page on Inverse problems.

--Dmharvey 13:56, 27 May 2005 (UTC)

Existence of solutions[edit]

Existence of solutions. There are two ways to prove a polynomial equation has solutions. The fundamental theorem of algebra says that a polynomial over the complex plane C has at least one zero in C. There is also Hamburger's Theorem which says that any field F can be embedded in a larger field which contains a solution to any polynomial equation over F. The latter theorem is based on algebraic constructions, quite independent of complex numbers and radicals.

Scott Tillinghast, Houston TX 07:30, 23 February 2007 (UTC)

Recommendation[edit]

Would someone fix some of the radical symbols to be done with TeX? It is currently exceedingly difficult to read.

Agree, for displayed equations at least. In the present state of mediawiki software, the inline radicals should be left alone. Dmharvey Talk 8 July 2005 16:53 (UTC)

Algebraic Equation[edit]

The first example says "Furthermore, it is true, but far less obvious, that this holds for every possible algebraic equation satisfied by A and B." And I thought, this clearly isn't true for the equation "A * A + B = c". But then I realised that that equation isn't algebraic because c contains sqrt(3). Anyway, my point is that the example is confusing to the non-mathematician who doesn't know the technical definition of algebraic equation. Algebraic equation currently redirects to Algebraic Geometry, which also isn't very helpful because it talks about Algebraic equations without defining them. I think wikipedia needs a simple article defining algebraic equations with links to and from this article and algebraic geometry. But I'm not qualified to do it. Reilly 15:02, 13 October 2005 (UTC)

On a related subject the piece currently reads:

'One might raise the objection that A and B are related by yet another algebraic equation,

 A-B-2\sqrt{3} = 0  ,

which does not remain true when A and B are exchanged. However, this equation does not concern us, because it does not have rational coefficients; in particular, √3 is not rational.'

But this last sentence cannot be correct can it?

a) Can galois theory really not handle complex coefficients?

b) isn't the real reason that that equation isn't part of the Galois group since it cannot be permuted?

WolfKeeper 19:05, 2 May 2006 (UTC)

I've had a stab at this: Algebraic equation

Reilly 17:55, 5 July 2006 (UTC)

Consider reading the article more than once.

"An important proviso is that we restrict ourselves to algebraic equations whose coefficients are rational numbers. (One might instead specify a certain field in which the coefficients should lie, but for the simple examples below, we will restrict ourselves to the field of rational numbers.)"

Consider F=Z[x]/(x^2+1). There are two automorphisms identity, and a+bx->a-bx. Here we are working over the field of integers. The equation x^2-1=0 has two solutions A=i and B=-i. However the equation A-B=2i implies that there is only the identity automorphism (since A and B cannot be interchanged). This is because the equation A-B=2i has coefficents in F which are not in the integers.

-kaz

Kaz, I am not satisfied with your answer. The link by Reilly also does not help. I support the claim by WolfKeeper. In which sense it "does not remain true when A and B are exchanged". What does in mean, "to remain true", while the equation has no solution among pairs of rational numbers? It cannot "remain true" because, for any rational A and B, it was not true. This example is not clear, we should either rewrite or remove it. dima (talk) 11:41, 11 October 2008 (UTC)
P.S. Is it possible to suggest some example, where  A-B appear (disabeling the swapping), but still remain in the set of rational numbers? dima (talk) 11:50, 11 October 2008 (UTC)

Inverse Galois problem[edit]

Someone has said that it is easy to construct field extensions with any given finite group as Galois group. That may be the case in algebraic function theory, but not when the ground field is Q. The problem was unsolved as of 1996, and I cite the book by Vōlklein. There is also a book (c 1997) by Malle and Matzat which reviews the history of the problem. Scott Tillinghast, Houston TX 05:14, 7 February 2007 (UTC)

The issue here is whether the base field is given or not. You can find some pair L/K with given G as Galois group; fix K and G and ask for L, and you have a hard problem. Charles Matthews 18:11, 7 February 2007 (UTC)

Applications[edit]

Could someone list or describe some of the applications of Galois theory, aside from proving there is no quintic equation? Sorry, that one just doesn't seem very exciting to me. What does one learn after Galois theory? Where do you go from here? What more advanced subjects use Galois theory? Do physicists or geometers ever have any use for Galois theory? What areas of active research use Galois theory?

A great thing to talk about would be its applications in Coding theory and computer science as well as its use in common CD-ROMs as well as things like WiMax. I added a link to RS ECC as an attempt to get this started. fintler (talk) 20:05, 7 January 2008 (UTC)

Fixed Field Redirect[edit]

Fixed field redirects here, but the article doesn't really explain the topic well. I suggest either changing fixed field back to a redlink or defining it in this page. 67.42.242.214 04:16, 24 October 2007 (UTC)

MS Word and capitalization[edit]

Micro$oft Word says that you should write Galois theory like "Galois Theory", why is that? 78.37.8.167 (talk) 19:31, 17 December 2007 (UTC).

Ask Microsoft... 129.241.211.33 (talk) 13:56, 24 May 2008 (UTC)

more precise statement[edit]

in the phrase: "the group of field automorphisms of L /K" means the automorphisms of L that preserve K? If so, isn't it better to spell this out? the pointer to the section on automorphisms doesn't clear this problem. —Preceding unsigned comment added by 155.198.157.118 (talk) 21:13, 16 September 2008 (UTC)

Yes, good idea. Go ahead! Be bold. Jakob.scholbach (talk) 13:35, 13 October 2008 (UTC)

Looks awful on a Macintosh computer[edit]

This article really looks awful on a Mac, Mac OS X, Version 10.5.5

Lots failed to parse. But looks great on a Sun Ray (at Sun Microsystems as I write)!

dino (talk) 21:30, 14 January 2009 (UTC)

Now looks fine on a Mac[edit]

Now looks OK on a Mac. What it was I don't know -- maybe a temporary local weirdness.

dino (talk) 22:21, 15 January 2009 (UTC)

Solvable[edit]

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)

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)

Discriminant is not a symmetric function in the coefficients[edit]

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)

You're not missing anything, it was indeed wrong. — Emil J. 14:06, 4 January 2010 (UTC)

solvable groups and solution by radicals not clear[edit]

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)

The great triumph[edit]

The article says "One of the great triumphs of Galois Theory was the proof that for every n > 4, there exist polynomials of degree n which are not solvable by radicals—the Abel–Ruffini theorem. This is due to the fact that for n > 4 the symmetric group Sn contains a simple, non-cyclic, normal subgroup, namely An."

But the criterion for solvability is that the factor groups of the normal composition chain are cyclic, not that the normal subgroup in the chains themselves are cyclic.

According to me the real reason why Sn is unsolvable, is because its subgroup An is unsolvable, because that one has no normal subgroups anymore which could lead to cyclic factor groups and the chain ends right there without resolving to {1}.

It's been too long since I've studied group theory so perhaps there is equivalence here, but for the Galois layman with some undergraduate math skills this argument is not intelligible I'm afraid. — Preceding unsigned comment added by 194.78.35.195 (talk) 10:49, 27 February 2013 (UTC)

The fact that An is simple and normal implies, by the definition of simple that (1) ⊂ An ⊂ Sn is a composition series. The fact that it is simple and non-cyclic implies that it is not solvable. Thus the formulation of the article and yours are both correct.
However, there is something else that is wrong in this paragraph: The "great triumph" of Galois is not that one, which was not a surprise after Abel-Ruffini result. It is that Galois gave an algorithmic criterion to decide which equations are solvable or not. This is better explained at the end of section "History". D.Lazard (talk) 12:17, 27 February 2013 (UTC)