I attended this talk, and it was one of the highlights of the conference for me. Mike is a great speaker and I found myself inspired to dive deeper into LLVM internals afterwards.
> I'm sure you can derive all sorts of weird metrics so that various weird identities are true
On the rational numbers, at least, the p-adic metrics are more or less your whole lot, according to Ostrowski's Theorem [1].
There is a kind of cognitive hurdle everyone who studies these numbers has to clear, in that things that should be "large" turn out to be very small indeed, when viewed under a p-adic lens. I think it's more instructive to build up the ring of p-adic integers first [2, chapter 2], and construct the p-adic numbers from there. I can assure you they are very useful, though! A general theme in number theory is to take a "global" problem, defined over the integers, and to translate it into infinitely many "local" ones (over the p-adics, for each prime p). These are sometimes easier to solve and, if you're lucky, offer insight into the global solution you're looking for.
It's terse but comprehensive, and the exercises have been a pleasant source of a-ha moments. There's a more recent version of the course offered, but I haven't been through it.
I really like GMB's approach to fitness (emphasis on movement and skill over reps and kilograms lifted) and have paid for a number of their programs. They know what they're talking about, and are incredibly humble about it too. Highly recommended.
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We already have a successful Android product, and our iPhone version will be launching within the week. We are looking for an experienced iOS developer to join our small team.
In 2009 during my Masters I was taking a course in elliptic curves and I was having a tough time getting to grips with them. I discovered Sage and was suddenly (after an eight-hour compile time!) able to easily create and manipulate these objects in an interactive environment. It was mind-blowing! I ended up doing quite well in that module and I'd say that's largely due to William Stein's work.
I'm no longer in academia and haven't used Sage for years but it's great to see how far this project has come. I hope it gets the funding and development it needs.
I took a History of Mathematics module as an undergrad and remember our lecturer telling us the main reason "New Math" never stuck was because parents were no longer able to help their kids with their homework.
I read quite a lot about Noether when I took a module in the history of mathematics as an undergraduate. Few mathematicians excel so greatly as to have an entire class of ring[0] named after them!
Modulo a prime ideal you're only guaranteed an integral domain. The residue ring is only a field when the ideal you're taking as a modulus is maximal. All maximal ideals are prime, but the converse is not true - for example in the ring Z[x] of polynomials with integral coefficients the ideal (p) generated by a rational prime p is prime but not maximal.
There are of course many rings in which all prime ideals are maximal. The integers, for example, and more generally the integer ring of any number field (which are not always principal ideal domains, but are examples of Dedekind domains).
