Since my last post I’ve started attending a course on-adic numbers. Initially my only real motivation for doing so was that a closely related concept had come up in my research; I had previously been of the opinion that the study of-adic numbers was something of a niche pursuit that bore little relevance to other areas of mathematics. However, having attended 2 lectures, I am finding the subject quite fascinating, and pleasing in the way it relates concepts from algebra, number theory and analysis. So today I’m going to write highly non-rigorously about some of the interesting bits…perhaps I will even do a short series of posts on the subject.
So what are the p-adic numbers? I think the best way to explain this is to start by talking about something a bit more familiar: the real numbers. A space is complete if, intuitively, it “has no gaps”; this is a very desirable property from the analyst’s point of view (in fact analysis can only be done in a complete space, as the notion of a limit does not make sense if there are gaps in the space). The formal definition of a complete space is one in which every Cauchy sequence – that is one in which the gaps between elements eventually get infinitesimally small – converges to a point in the space. The rational numbers are not complete because, for example, we can construct a sequencethat converges toby defining:
…and so on. The real numberscan be obtained by completing the rational numbers, that is, by “filling in the gaps”. The way we do this is to take every Cauchy sequence inand letbe the set of points that these sequences converge to (for the more technically-minded, is the quotient ring, whereis the ring of Cauchy sequences inandis the maximal ideal ofconsisting of all sequences converging to zero). A helpful way to think of this is by envisaging the decimal expansion of every number as being a convergent sequence, in the same way as we saw above for. Sequences are considered to be equivalent if they converge to the same point, and so for example, because the sequences:andboth converge to