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

towardsthe 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.

I say:

“Tell us what the main theorem is, give us an idea of how it works, and

thengo 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

are.

Let’s put this into context. You will remember from school that the two solutions to the equation

are:

and .

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 …

Please carry on your explaination on this tough topic . I think that you are achieving that nobody did

I have always wanted to know about galois theory and this is the most easy to understand I have seen so far. Please carry on,

Will do!

Robert,

Thank you for the illuminating presentation.

Looking forward to the next installment of GTFD.

Nice so far. Very clear. Why aren’t you working on this 24 x 7? Joking only.

So is there some theory that defines what a normal extension is independently of containing all the roots of a related polynomial? Which polynomial?

Or is that all there is. Actually I am prepared to suspend this question and come back to it if it is a big diversion. I think you have given me the main theme of what a normal field extension, without the detail. Which is good. It’s not completely satisfying but it is better than reading a 200 page book 😉

But please carry on.

Excellent article!

In the update, the first link to an article that covers 80% of the right kind explanation is missing. Could you please add that link?

Thanks, again. Very intuitive article.

Awesome! Can’t wait to see the next one

Great explanation so far. Would be great if you had the time to finish on these. Would you be able to list the next steps in understanding galois theory? what would come after step 2 irreducible polynomials? Would be really helpful to know. Thank you

Norman Wildberger also did a good youtube on it with symmetric polynomials.

This one I think: https://www.youtube.com/watch?v=cxNq-hQwvn0

Oh, also check out by Boaz Kurtz, a video showing the commutator: https://www.youtube.com/watch?v=RhpVSV6iCko&list=LLMJ9FgxJhTuBrU9cIH-LmBg&index=43

(afaik this is exposition is due to V I Arnold)

Galois Theory for Beginners ; John Stillwell

The American Mathematical Monthly

Vol. 101, No. 1 (Jan., 1994), pp. 22-27

The intro to this article states that one doesnt need normal field extension, or Galois correspondences between subfields and subgroups to prove the unsolvability of quintics. It goes on to list the three key ingredients required for this proof, and claims that 90% of what is taught in standard treatments is unnecessary to get to the result.

Thanks for posting. I am a huge fan of John Stillwell’s book ‘Mathematics and its History’. I am looking forward to reading this article you recommend.

Thanks for your posts. I found this link is a good site to get an idea of the basics.

https://math.stackexchange.com/questions/62027/intuition-behind-looking-at-permutations-of-the-roots-in-galois-theory

Thank you very much for posting the link, very nice read.