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The ring of p-adic integers

By ellipticcurves

In what way can we impose operations of addition, subtraction, and multiplication on the set of p-adic integers? One thing on our wish list is that these operation be compatible with the corresponding operations on the ring of integers embedded in {\mathbb Z}_p. Another, is that they be compatible with the operation on the component considered as elements of {\mathbb Z}/p{\mathbb Z}. It seems that the obvious way to define these operations is componentwise:

  • (u_1, u_2, \ldots) + (v_1, v_2, \ldots) := (u_1 + v_1, u_2 + v_2, \ldots);
  • (u_1, u_2, \ldots) - (v_1, v_2, \ldots) := (u_1 - v_1, u_2 - v_2, \ldots);
  • (u_1, u_2, \ldots) \cdot (v_1, v_2, \ldots) := (u_1 v_1, u_2 v_2, \ldots).

Another way to define these operations is using the “series” notation. We perform these operations just like base p calculations, only digits extend infinitely to the left (some people write \ldots b_2 b_1 b_0 instead of the series notation b_0 + b_1 p + b_2 p^2 + \cdots). For example, (2 + 1 \cdot 5 + 3 \cdot 5^2 + \cdots)(1 + 2 \cdot 5 + 4 \cdot 5^2 + \cdots) = 2\cdot 1 + (2\cdot 2 + 1\cdot 1) 5 + (2\cdot 4 + 1\cdot 2 + 3\cdot 1) 5^2 + \cdots = 2 + 0\cdot 5 + 4\cdot 5^2 + \cdots. We carry 1 toward 5^2’s and 2 towards 5^3’s. It’s a bit of an exercise to show that these definitions agree. It is not that difficult to check that these operations make {\mathbb Z}_p into a commutative ring.

Using Abstract Nonsense, the construction of the p-adic numbers as sequences can be reformulated as follows. For each r we have a map

{\mathbb Z}/p^{r+1}{\mathbb Z} \to {\mathbb Z}/p^r{\mathbb Z}

of reduction modulo p^r, so rings {\mathbb Z}/p^r{\mathbb Z} form an inverse system. Then

{\mathbb Z}_p = \displaystyle\mathop{\mathrm{lim}}_{\longleftarrow}\ {\mathbb Z}/p^r{\mathbb Z},

the inverse limit of this system. Recall that the inverse limit in the category of rings is constructed by taking all elements of the direct product of all rings in the system (a.k.a. sequences) that “agree” with the reduction maps.

This entry was posted on December 26, 2008 at 2:20 am and is filed under 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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