Talk:Galois theory/Archive 1

Archive 1

Compass and straightedge

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, 17 October 2002 (UTC)

Automorphism group versus Galois group

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 February 2004 (UTC)

Bold changes

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)

Recommendation

Would someone fix some of the radical symbols to be done with TeX? It is currently exceedingly difficult to read. — Preceding unsigned comment added by Guardian of Light (talk • contribs) 16:28, 8 July 2005 (UTC)

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)

Inverse Galois problem

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)

Existence of solutions

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)

Fixed Field Redirect

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)

Applications

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? — Preceding unsigned comment added by Singularitarian (talk • contribs) 17:33, 9 August 2007 (UTC)

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)

MS Word and capitalization

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

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

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

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)