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

No.04:

In this lecture, rings are assumed to be unital, associative and commutative unless otherwise specified.

DEFINITION 04.1   A (unital commutative) ring $A$ is said to be a local ring if it has only one maximal ideal.

LEMMA 04.2   Let $A$ be a ring. Then the following conditions are equivalent:

1. $A$ is a local ring.
2. $A\setminus A^\times$ forms an ideal of $A$.

PROPOSITION 04.3   $\mathbb{Z}_p$ is a local ring. Its maximal ideal is equal to $p \mathbb{Z}_p$.

We may do some “analysis” such as Newton's method to obtain some solution to algebraic equations.

Newton's method for approximating a solution of algebraic equation.

Let us solve an equation

$$x^2=2$$

in $\mathbb{Z}_7$. We first note that

$$3^2\equiv 2\quad (7)$$

hold. So let us put $x_0=3=[0.3]_7$ as the first approximation of the solution. The Newton method tells us that for an approximation $x$ of the equation $x^2=2$, a number $x'$ calculated as

$$x'=\frac{1}{2}(x+ \frac{2}{x})$$

gives a better approximation.

$$x_0'=\frac{1}{2}([0.3]_7+[0.3\dot{2}]_7=[0.3\dot{1}]_7$$

So $[0.3\dot{1}]_7$ is a better approximation of the solution. In order to make the calculation easier, let us choose $x_1=[0.31]_7$ (insted of $x_0'$) as a second approximation.

$$x_1' =\frac{1}{2}([0.31]_7+2/[0.31]_7) =\frac{1}{2}([0.31]_7+[0.3\dot{1}45\dot{2}]_7)\fallingdotseq [0.312]_7$$

We choose $x_2=[0.312]_7$ as a second approximation.

$$x_2' =\frac{1}{2}([0.312]_7+2/[0.3\dot{1}2534066\dot{2}]_7)\fallingdotseq [0.31261]_7$$

We choose $x_3=[0.31261]_7$ as a third approximation.

$$x_3' = =\frac{1}{2}([0.31261]_7+[0.3126142465066\dots]_7)\fallingdotseq [0.312612124...]_7$$

We choose $x_4=[0.312612124]_7$ as a third approximation.

$$x_4' =\frac{1}{2}([0.312612124]_7+[0.312612124565220422662213135351\dots]_7)$$

$$\fallingdotseq [0.3126121246621102]_7$$

EXERCISE 04.1   Compute $[0.5]_7/[0.11]_7$

EXERCISE 04.2   Find a solution to

$$x^3\equiv 5 \pmod {11^5}$$

such that $x \equiv 3 \pmod {11}$.

DEFINITION 04.4   We denote by $\mathbb{Q}_p$ the quotient field of $\mathbb{Z}_p$.

LEMMA 04.5   Every non zero element $x \in\mathbb{Q}_p$ is uniquely expressed as

$$x= p^k u \qquad( k\in \mathbb{Z}, u \in \mathbb{Q}_p^\times).$$

We have so far constructed a ring $\mathbb{Z}_p$ and a field $\mathbb{Q}_p$ for each prime $p$.

PROPOSITION 04.6   Let $p$ be a prime. Then:

1. $\mathbb{Z}_p$ is a local ring with the unique maximal ideal $p \mathbb{Z}_p$.

2. $$\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p (=\mathbb{Z}/p \mathbb{Z}).$$

3. $\mathbb{Z}_p$ is an integral domain whose quotient field $\mathbb{Q}_p$ is a field of characteristic zero.

With $\mathbb{Q}_p$ and/or $\mathbb{Z}_p$, we may do some “calculus” such as:

THEOREM 04.7   [#!Serre2!#, corollary 1 of theorem 1] Let $f\in \mathbb{Z}_p[X_1,X_2,\dots,X_m], x\in \mathbb{Z}_p^m$ , $n,k\in \mathbb{Z}$. Assume that there exists a natural number $j$ such that $1\leq j \leq m$,

$$\frac{\partial f}{\partial X_j} (x)\not \equiv 0 \pmod p.$$

Then there exists $y\in \mathbb{Z}_p^m$ such that

$$f(y)=0$$ (1)

$$y\equiv x \pmod p$$ (2)

See [#!Serre2!#] for details.