Wednesday, 8 June 2011

Galois Theory Lie Groups and distractions

Here I am 1 week and 2 weeks away from TMA deadlines and I should be getting my head down trying to polish them off. However I got distracted a couple of weeks ago by a link someone posted on one of the forums to a book on Lie Groups Physics and Geometry, Chapters of which are accessible here.

The first chapter in particular caught my attention as it gave an accessible account of Galois Theory. As most reading this blog will know Galois theory arose out of the attempt to find a formula similar to those available for quadratic equations, cubic equations and quartic equations. It was one of the many strands leading into our current group theory. Most books on abstract algebra eg Artin leave Galois theory till the end with the implication that you have to master all sorts of arcane subjects such as rings, fields, field extensions etc before you can get the gist of what is going on. Of course Galois himself was unaware of such theories and it is doubtful if he could pass an exam in a subject of which he is the founder. It is refreshing to find an account which tells you the point before launching into theorem after theorem. Of course once one has read such a chapter one can then go onto the more formal stuff. I was slightly distracted and probably missed the point by a transformation which applies to the quadratic equation

$$x^2+px+c=0$$ if we make the transformation x -> -x-p then a quick substitution shows that the equation doesn't change. I had hoped that one could find global invariants of the cubic equation


which if we write as

$$(x-r1)(x-r2)(x-r3)=0$$  where r1,r2 and r3 are the three roots of the equation  we can obtain expressions for p,q and c in terms of the roots Namely.

c =  -(r1 r2 r3)
p =  -(r1+r2+r3)
q =  r1r2 + r1r3 +r2r3

So you might think that you should be able to find a transformation similar to the quadratic equation transformation which leaves the equation invaraint. I thought by expanding two of the brackets in the root equation eg
$$(x-r1)(x^2-(r2+r3)x-r2r3)$$ that you could apply the quadratic transformation

x -> -x - (r2+r3)
and lo and behold recover the original cubic equation. I wasted hours trying to find invariants but not for the first time in my life I was up the garden path. because of course you have transform the other bracket so x-r1 -> -x-(r1+r2+r3) which when expanded does not lead you back to the original cubic. However it was fun whilst the illusion lasted I felt alive and joyful and it was a refreshing change to do some maths for its own sake not simply because yet another TMA question had to be answered.  A bit of further reading showed me that Galois was concerned with symmetries between the roots rather than global symmetries of the equation itself. From the above expressions for p.q and c all sorts of invariants can be constructed also note that permuting the roots leaves the above expressions invariants. Thus there is a close connection between group theory and something as mundane as solving polynomial equations.
My experience was a  lesson in being overoptimistic, just as a bad chess player (which I am) will spend hours developing an attack, only to miss the obvious fact that his queen is under threat and the priority should be to move the queen out of danger. Just as my priority should be to finish my two TMA's. I take heart that my mate Neil who is a good chess player and is revising  for a computing exam had a similar distraction

Anyway along with all the other distractions I've got an interest in Galois theory and have ordered this book by Cox which  Nilo recommends

It seems accessible in that it starts from the motivation of the solution of quadratic, cubic and quartic equations

Also a good set of notes available on the net amongst many is here

So yet another distraction but one thats fun hopefully I've got it out of my system and can concentrate on the TMA's over the next two weeks.


  1. Actually Galois theory started with Lagrange.He developed so many transformations you were talking about. And it is always so much fun to go through a historical approach while learning Galois theory(Also number theory and calculus). To see how Lagrange tried to attack this problem look at Dickson's "Algebraic theories" book.

  2. Thanks for the links! Have you tried

  3. Hi thanks for commenting No I haven't looked at that book.
    Yet another one ;) to add the list.
    Unfortunately my enthusiasm for Galois theory has waned a bit since the nightmare of my Topology course. When the nightmare has finished hopefully I'll get back into it.
    There is a little book

    Which should provide a good entry point before tackling a more formal book as Cox. When eventually the new Open University Further maths course Further pure maths M303 comes on the scene there are some units on Fields and rings so I will probably do that course. It's not due for another couple of years yet