Talk:Galois theory

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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)

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)

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)

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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. 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)

[edit] Recommendation

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)

[edit] Algebraic Equation

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√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

[edit] 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)