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
Now, to say a sequence converges to a point is to say that it gets “infinitely close” to that point, and so the notion of convergence depends on what we mean by “close”. There are many different metrics – that is, ways to measure distance – in mathematics. In the above construction of the real numbers the metric we used was induced by the usual absolute valueso we considered the distancebetween two numbersandto beiforif
Letbe some prime, and let
be the prime decomposition of some rational number, where theare distinct prime numbers different fromand theare integers. We then define the-adic absolute valueof to be . So for example, the-adic absolute value ofis, because the prime decomposition ofis, andfor any other prime
You will no doubt have noticed that numbers which are divisible by higher powers ofhave a smaller-adic absolute value. Using this notion of size to measure the space between numbers provides us with another, rather counterintuitive way to define distance, in which numbers which differ by a large power ofare thought of as being close together. For example, in the-adic metric the distance betweenandiswhile the distance betweenandis. This means that Cauchy sequences in the-adic metric look rather different from those we are used to. Take the sequence:
In the-adic metric, the gap between theth andth element is:
so these get smaller and smaller: this is a Cauchy sequence. In fact, this sequence converges to zero. But just as was the case with the usual absolute value metric, there are Cauchy sequences which do not converge in the-adic metric, eg:
Again the sizes of the gaps between elements here approach zero. But the limit of this sequence is:
which is not a rational number. By adding the limits of sequences like this one in the same way described above for the real numbers, we can complete the rationals in a different way: this time with respect to the-adic metric. This gives us the field of-adic numbers,Because is complete we can do–adic analysis on it, in much the same way as we do real analysis onand complex analysis onBut why would we want to do that? And really, what is the point of any of this? Hopefully I will get to that next time….