September 12th, 2010


Project "learn Galois theory" : status report


As usual, I started out full of hope. Abstract algebra is deceptively easy in the beginning-- everyone can understand groups, cos they come up all the time in real life: clock arithmetic, rotations of a cube, etc. So you breeze through confidently. But the thing about math textbooks is, every single sentence matters. If you skip one proposition or gloss over one definition, it comes back to bite you in the ass 50 pages later. There's no royal road to geometry, or however that goes.

So of course my progress got slower and slower as the text moved from the easy going terrain of groups to the rockier hills of rings and fields. Last time I got stuck at cosets & "normal" subgroups. This time I studied them carefully and got through, only to run into a fucking mountain range of "ideals" and "quotient rings" and "lattices of automorphisms".

fig a: WTF

What the hell IS an ideal? There's examples in the text but it hurts my brain as soon as you move away from integers. An ideal is a subset I of a ring R, closed under subtraction/addition in I, closed under multiplication for R*I (any element in R * any element in I = element in I) ... it's involved in 'quotient rings'... I don't know. Why's it even called an ideal?! I really have no idea.

Here are two examples I KIND OF get:
1) {0} is an ideal in Z (ring of integers).
2) "multiples of n" is an ideal in Z. the ideal generated by n is written as <n>, e.g. <3> = {+-3, 6, 9, 12, 15...}

Given a ring R and an ideal I within it, the 'quotient ring' R/I is the set of equivalence classes (cosets) for R in I. It's like how addition/multiplication mod 3 partitions the integers into 3 equivalence classes-- 0, 1, 2. The quotient ring Z/<n> is isomorphic to the ring for addition/multiplication in Z mod n. I think.

So ok I sort of get that, but then they start talking about quotient ring where the top part is a field of *polynomials*, F[x]. and the bottom part is the ideal generated by a polynomial. e.g. "F[x] / <x^2 + 1>" WTF? What the hell does that quotient ring look like? What are the equivalence classes for addition/multiplication in a field of polynomials, modulo a polynomial?

By now I'm sobbing quietly in my chair...

Fuck this, I'ma take up gardening or glue sniffing, either would be a better use of my time