Hardy and Wright, Chapters 12 and 13

Guess I kept getting distracted from posting about our meeting last week about Chapter 12. So here it is (what I remember of it – that was a long time ago), with notes from today’s meeting as well.

Honestly, we didn’t talk too much about the content of chapter 12, directly. The three of us were fairly comfortable with most of it from our algebra classes a few years ago. But Chris asked “How many of these quadratic extensions are Euclidean domains?” He had looked into this before the meeting, and found that it is an active topic of research, and that some results are known. I pointed out that we’d get to see some of them in Chapter 14.

Eric mentioned that the norm N(a+b\sqrt{m})=a^2-mb^2 could be attained in a more general fashion. Namely, these sets of “numbers” are elements of degree 2 extensions of \mathbb{Q}. That means they are a 2 dimensional vector space over \mathbb{Q}. And each element a+b\sqrt{m} acts on that vector space as a linear transformation. By writing down a matrix for that transformation and taking it’s determinant, you recover the norm. He thought these ideas had further generalizations, but I don’t remember how much he told us about.

I do remember him telling us that this result that 1-\rho is a prime (\rho being the primitive third root of unity, prime in what HW denote k(\rho)) was also somewhat generalizable. Somehow, 1-(thing) is often a prime, or so he said. I tried thinking about 1-\rho in the complex plane, but had no idea what the geometry would tell me about it being prime, or vice-versa. Eric and I talked about it and the relation to the hexagonal lattice, but didn’t get too far.

So that was what our talk inspired by Chapter 12 covered (at least, that’s what I remember of it). Today, Chris and I met and talked briefly about Chapter 13. Again, neither of us had too much to say specifically about the reading. I mentioned that I had just used the results about all the Pythagorean triples to solve a Project Euler problem.

From the chapter notes, I saw that the result about x^3+y^3+z^3=0 having no integer solutions was given as an exercise in something by Landau. I asked Chris if he knew how to phrase any of it in terms of ideals, since he’s an algebraist, but he didn’t. I also asked if some of the manipulations in 13.6, specifically the dividing by Z at some point, was inspired by some sort of projective space to affine space conversion. Dividing by capitals Zs apparently makes me think that. Chris allowed how it might be the case, and that algebraic geometry probably could come in here. But neither of us really had much specific to say.

Finally, I relayed the historical anecdote about the integers that are the sum of two cubes in two different ways. The story about Hardy visiting Ramanujan in the hospital, saying that his cab number (1729) wasn’t particularly interesting, and Ramanujan pointing out that it was the smallest integer expressible as the sum of two (positive) cubes in two distinct ways. I didn’t relay how I remembered noticing that 1729 was one of the house numbers in the movie Untraceable. I don’t know if they did that on purpose or not, but it’s there.

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