Automorphisms of the Splitting Field

I learned something over the last few days that has made a big difference in the way I understand Galois theory. I told you the other day that my friend in West Bengal had shown me how to generate functions of the five roots of the quintic which were themselves solutions of a sixth-degree equation, the so-called "resolvent". (It seems the term "resolvent" is used to describe both the functions of the roots and the resulting equation.) In the course of our follow-up discussions, the question arose as to the Galois Group of x^4-2=0....that is, the allowable permutations of the fourth roots of two.

The Galois Group describes the set of automorphisms of the splitting field of a polynomial, and it is well know that any automorphism must map roots of the polynomial to each other. So people normally describe the automorphisms in terms of where they send the roots. My correspondent made the observation that in the present case, the mappings seemed to correspond to flips and rotations of the four points on the circle whose radius was the fourth root of two...in other words, the dihedral group of four elements, or D4.

What I realized is that while all this is technically correct if we just restrict ourselves to looking at the fourth roots of two, it gives a very deceptive picture of what the field automorphisms really look like. Mathematicians are always overly proud of economy of analysis, so having proven that an automorphisms is fully defined by its action on the roots, they are happy to ignore everything else that is going on. And in this case, there is indeed much more going on.

It is true that the splitting field of the polynomial is generated by the fourth roots of two, which are located 90 degrees apart on the circle in the complex plane of that radius. But this same field also includes two more bigger circles that will be of great interest...a second circle containing the fourth roots of four (including the ordinary square roots of two), and third circle containing the fourth roots of eight.

The funny thing about automorphisms of the field is that if you describe them in geometric terms, like flips and rotations, it turns out these three rings behave very differently. The automorphism which rotates the inner ring clockwise simultaneously rotates the outer ring counterclockwise! And the middle ring gets flipped horizontally and vertically. In other words, the complex plane is not smoothly transformed, but rather it is completely scrambled...as it must be. That is the nature of pure algebra...the proximity between two points has absolutely no significance. How can it if rational points are always mapped to each other, while irrational points are swapped about?

So I worked out all the automorphisms of the fourth roots of two, keeping track of all three "rings" of roots instead of focussing just on the inner ring.  It completely changed my perspective on the nature of automorphisms. For example, I have always known that the cube roots of two are theoretically indistinguishable in Galois theory. Obviously the two complex roots can be swapped with each other...I had no problem with that. But why are you allowed to pick one of those complex roots and swap it with the real root? Doesn't that just seem wrong? And the answer is...not when you simultaneously look at what the cube roots of four  are doing at the same time. It turns out that while the real cube root of two is being swapped with its positive complex conjugate, the cube root of four is being swapped with its negative conjugate. So the overall symmetry of the complex plane is somehow restored.

When I wrote last year about the Galois Group of {x^5-2 = 0}, I laboriously tracked the logic of where the roots had go assuming you started by rotating them counter-clockwise. What I didn't realize was the bigger, much more beautiful picture. There are actually four concentric rings worth looking at...the fifth roots of two, four, eight, and sixteen. The automorphism which rotates the inner ring counter-clockwise simultaneously rotates the next ring counter-clockwise by two notches, and so forth, so the outer ring is actually rotated clockwise. There is a complete symmetry! It gets much more intricate when you start considering the allowable permutations between the fifth roots of unity...when you mix these in you ultimately get what is called the Frobenius Group, a set of twenty automorphisms in all. But all those re-shufflings, which appear so arbitrary and downright cock-eyed when you only look at the fifth roots of two, become so much more aesthetically satisfying when you consider what all four rings are doing simultaneously.