January 29, 2010

Since my last post I’ve started attending a course on$p$-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$p$-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 sequence$(a_i)$that converges to$\sqrt{2}$by defining:

$a_1=1$

$a_2=1.4$

$a_3=1.41$

$a_4=1.414$

…and so on.  The real numbers$\mathbb{R}$can be obtained by completing the rational numbers$\mathbb{Q}$, that is, by “filling in the gaps”.  The way we do this is to take every Cauchy sequence in$\mathbb{Q}$and let$\mathbb{R}$be the set of points that these sequences converge to (for the more technically-minded,$\mathbb{R}$ is the quotient ring$\frac{C}{M}$, where$C$is the ring of Cauchy sequences in$\mathbb{Q}$and$M$is the maximal ideal of$C$consisting 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$\sqrt{2}$.  Sequences are considered to be equivalent if they converge to the same point, and so for example$0.9999...=1.000...$, because the sequences:$0,0.9, 0.99, 0.999,...$and$1, 1.0, 1.00 ,1.000,...$both converge to$1.$