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Teichmüller Representatives

By ellipticcurves

Let us solve a particular equation in p-adic numbers:

x^p - x = 0.

Notice that since this equation has precisely p distinct solutions in {\mathbb F}_p={\mathbb Z}/p {\mathbb Z} (actually, all of its elements), by p-adic mojo, it has p solutions in {\mathbb Z}_p.

It is easy to figure out the sequence representations of these solutions:

(m, m^p, m^{p^2}, \ldots)

for m=0,\ldots, p-1. By Euler’s Theorem,

m^{p^r} \equiv m^{p^{r-1}} \pmod{p^{r-1}},

which takes care of both the condition on sequences defining a p-adic number and the fact that every element of the sequence is a solution of our equation in its respective modular ring.

Teichmüller’s insight was that it might be easier to write down the arithmetic laws on p-adic numbers expressed as p-series in these numbers (zero and (p-1)st roots of unity) rather than just the integers 0, \ldots, p-1 themselves. But more on that later.

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This entry was posted on January 21, 2009 at 1:00 am and is filed under Witt vectors, p-adic numbers. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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