This post is a LeanPost: it will be developed further depending on feedback from my readers.
See my note here on what a LeanPost is.
Update, January 2014: Dear Reader, for the last year I have been struggling to find time to work on this blog, and this article is particularly in need of further work. The start is good, and people have commented that they like it, but for me to continue I need to devote quite a lot of energy to my Galois notes and I think this is not going to happen soon (I am very busy at PrismFP you see). In the meantime, let me just add one link to an article which I think is about 80% the right kind of explanation of Galois theory. Here it is (in my opinion it could be improved, but I cannot do that at the moment so I leave you only with the link). Here is another presentation which is good too. End of update.
The most popular undergraduate text (in the UK at least) for a course in Galois Theory is by Ian Stewart from the University of Warwick (see here). If you google for “Galois Theory filetype:pdf” there are a few presentations available. A couple that sit around the top of the search results are by:
- Brent Everitt from the Mathematics department at York University (see here),
- Miles Reid at Warwick too (see here).
My principle complaint at all these texts is that they make exactly the same mistake (though Miles Reid does try his best to avoid it):
they work towards the main result rather than starting from it.
That’s often the case for lots of mathematical texts, but for Galois Theory it is particularly frustrating because there is actually a central theorem.
“Tell us what the main theorem is, give us an idea of how it works, and then go through all the details of the proof”.
Not the other way around!
This post is my effort at An Introduction to Galois Theory that starts at the interesting bit.
The big question that Galois answered
Galois produced a framework which allows us to prove that we cannot ever find a formula that tells us what the zeroes of the polynomial
Let’s put this into context. You will remember from school that the two solutions to the equation
There you are: the solutions expressed as simple formulas based on the coefficients a, b and c.
Similar formulas exist for the cubic and the quartic polynomial equations, albeit a bit more complicated, so it won’t come as a surprise that lots of mathematicians invested a lot of effort to find such formulas for polynomials with an term, aka quintics.
Galois stood back and asked whether it might just be impossible. He then did his work and proved that it is impossible.
How did he do it?
The way that Galois showed that the quintic cannot be ‘solved’ was pretty clever, and here it is in simple bullet-point terms which we will expand as we go along:
- The formulas for the solutions of all look like towers: using square-roots, square-roots of square-roots, or cube-roots of square-roots, and so on upwards.
- Galois realized that the numbers that can be reached from these sorts of step-by-step towerings of square-roots or cube-roots or suchlike must all have a quite specific kind of symmetry.
- We have an example of a quintic polynomial whose solutions have a less rigid symmetry than that kind of specific symmetry which formulas generate.
- Therefore it must be impossible to represent its solutions in terms of those kinds of formulas.
Saying this again in different words: all these ‘solution formulas’ are built on top of the integers in a way which generates collections of numbers that have a specific kind of symmetry, and since we have an example of a polynomial in whose solutions do not contain this symmetry we must deduce that it is not solvable by a formula.
Remark: it’s the beginning of the end, not just a hiccup
According to Galois Theory, the reason that polynomials in do not have a general solution formula is because the group contains a non-abelian normal subgroup that does not have any non-trivial normal subgroups (it’s the subgroup ).
All I want to say at this point is that in some sense things work up to because the groups and are too small to have much room for anything to go wrong.
I am saying it in this way so that you get a sense that it is not so much that things were going well and then we just had a hiccup for , but rather the opposite: things went well for , and because they are special cases – the groups are too small to have any ‘bad’ behaviour.
Learning Group theory with Galois
Clearly Galois Theory depends a lot on the concept of symmetries: group theory, that is. I am assuming that you have done a course on group theory, and have met the concept of normal subgroups.
The great news is that Galois theory gives us a better intuition of what normal subrgoups are. I delve into that in another post.
Getting into the details. Step 1: field extensions
The textbooks on Galois theory quickly get up and running with the concept of field extensions. An example to bear in mind is something like : the set of rational numbers with added.
However, the fields that Galois was interested in for his framework have much more rigidity than a general field extension, so it’s a pity that these textbooks drag you through a study the general properties of field extensions – you won’t need them in order to understand Galois.
In the textbooks these special extensions get called normal. (In Ian Stewart’s book you don’t meet them until almost half way through.)
In a few words, normal field extensions are those which are generated by including all the zeroes of polynomials.
For example, is normal because it contains both the zeroes of the polynomial .
Another example, is not normal because the polynomial has another two zeroes which are not in there: namely and .
The main property of these normal extensions is that any irreducible polynomial which has a zero in the extension will have all of its zeroes in that extension.
We will look at this more carefully now, because it is telling us something important about the zeroes of irreducible polynomials:
the zeroes of irreducible polynomials cannot be separated from each other.
Getting into the details: Step 2: irreducible polynomials
To be continued …