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Motivation for Witt Vectors

By ellipticcurves

If we write two p-adic integers a and b as p-series in integers 0, \ldots, p-1, and then multiply them together to get c, it goes something like this:

(a_0 + a_1 p + \cdots)(b_0 + b_1 p + \cdots) = a_0 b_0 + (a_1 b_0 + a_0 b_1) p + \cdots

Therefore, c_0 \equiv a_0 b_0 \pmod{p}. So far so good. But as we move to the next congruence class, we get

c_1 \equiv a_1 b_0 + a_0 b_1 + \text{ carry from }a_0 b_0 \pmod{p}.

Good luck figuring out the formula for the carry part.

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This entry was posted on June 1, 2009 at 3:55 pm 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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