$\mathbb{Z}_p$, $\mathbb{Q}_p$, 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
   $$(x+y)^p=x^p +y^p$$

We would like to show existence of “finite fields”. A first thing to do is to know their basic properties.

LEMMA 0.3   Let $F$ be a finite field (that means, a field which has only a finite number of elements.) Then:

1. There exists a prime number $p$ such that $p=0$ holds in $F$.
2. $F$ contains $\mathbb{F}_p$ as a subfield.
3. $q=\char93 (F)$ is a power of $p$.
4. For any $x\in F$, we have $x^q-x=0$.
5. The multiplicative group $(F_q)^{\times}$ is a cyclic group of order $q-1$.

The next task is to construct such fields. An important tool is the following lemma.

LEMMA 0.4   For any field $K$ and for any non zero polynomial $f\in K[X]$, there exists a field $L$ containing $L$ such that $f$ is decomposed into linear factors in $L$.

To prove it we use the following lemma.

LEMMA 0.5   For any field $K$ and for any irreducible polynomial $f\in K[X]$ of degree $d>0$, we have the following.

1. $L=K[X]/(f(X))$ is a field.
2. Let $a$ be the class of $X$ in $L$. Then $a$ satisfies $f(a)=0$.

Then we have the following lemma.

LEMMA 0.6   Let $p$ be a prime number. Let $q=p^r$ be a power of $p$. Let $L$ be a field extension of $\mathbb{F}_p$ such that $X^q-X$ is decomposed into polynomials of degree $1$ in $L$. Then

1. $$L_1=\{x \in L; x^q=x\}$$
   is a subfield of $L$ containing $\mathbb{F}_p$.
2. $L_1$ has exactly $q$ elements.

Finally we have the following lemma.

LEMMA 0.7   Let $p$ be a prime number. Let $r$ be a positive integer. Let $q=p^r$. Then we have the following facts.

1. There exists a field which has exactly $q$ elements.
2. There exists an irreducible polynomial $f$ of degree $r$ over $\mathbb{F}_p$.
3. $X^q-X$ is divisible by the polynomial $f$ as above.
4. For any field $K$ which has exactly $q$-elements, there exists an element $a\in K$ such that $f(a)=0$.

In conclusion, we obtain:

THEOREM 0.8   For any power $q$ of $p$, there exists a field which has exactly $q$ elements. It is unique up to an isomorphism. (We denote it by $\mathbb{F}_q$.)