$\mathbb{Z}_p$, $\mathbb{Q}_p$, and the ring of Witt vectors

Yoshifumi Tsuchimoto

You should know that every positive integer may be written in decimal notation:

$$(531)_{10}=5\times 10^2 +3\times 10^1+1\times 10^0.$$

Similarly, given any integer (“base”) $b\geq 2$, we may write a number as a string of digits in base $n$. For example, we have

$$(531)_{10}=1\times 7^3+3\times 7^2 + 5 \times 7 +6 \times 1=(1356)_7.$$

Similarly, we have

$$(531)_{10}=(1356)_7=(1023)_8=1000010011_2=(213)_{16}.$$

You may also probably know (repeating) decimal expresions of positive rational numbers.

$$(531.79)_{10}=5\times 10^2 +3\times 10^1+1\times 10^0+ 7\times 10^{-1} +9\times 10^{-2}.$$

$$(531.79)_{10}=(1356.\dot{5}34\dot{6})_{7} =(1023.62\dot{4}365605075341217270\dot{2})_{8}$$

Now let us reverse the order of digits. Namely, we employ a notation like this¹:

$$[97.135]_{10}=(531.79)_{10}$$

$$[0.135]_{10}=(531)_{10}$$

$$[123.456]_{10}=(654.321)_{10}$$

$$\dots$$

Let us do some calculation with the above notation:

$$[0.1]_{10}+ [0.9]_{10}=[0.01]_{10}$$

$$[0.1]_{10}\times [0.9]_{10}=[0.9]_{10}$$

$$[0.01]_{10}\times [0.09]_{10}=[0.009]_{10}$$

You may recognize curious rules of computations. This curious notation will lead you to a new world called “the world of addic numbers”.

EXERCISE 0.1   Compute

$$[0.12345]_8+[0.75432]_8$$

with our curious notation. Then do the same computation in the usual digital notation in base $10$.

LEMMA 0.1   For any prime number $p$, $\mathbb{Z}/p \mathbb{Z}$ is a field. (We denote it by $\mathbb{F}_p$.)

LEMMA 0.2   Let $p$ be a prime number. Let $R$ be a commutative ring which contains $\mathbb{F}_p$ as a subring. Then we have the following facts.

1. $$\underbrace{1+1+\dots+1 }_{\text{$p$-times}}=0$$
   holds in $R$.
2. For any $x,y\in R$, we have
   $$(x+y)^p=x^p +y^p$$