: P-adic ultrametric<a class="btn btn-sm" href="/UUID/EDB/2CG/?lang=eng"><span class="ttfamily"><small class="scriptsize">[2CG]</small></span></a>

10.14 P-adic ultrametric[2CG]

We report from the notes [ 2 ] the definition of the $p$–adic distance on the $ℚ$ set. Let $p$ be a fixed prime number.

Definition 307

[0XF] Each rational number $x≠0$ breaks down uniquely as a product

\begin{equation} \label{primi} x=± p_ 1^{m_ 1}p_ 2^{m_ 2}\cdots p_ k^{m_ k}\ , \end{equation}

308

where $p_ 1{\lt}p_ 2{\lt}\cdots {\lt}p_ k$ are prime numbers and the $m_ j$ integers. Fixed as above a prime number $p$, we define the $p$–adic absolute value of $x∈ℚ$ as

\[ |x|_ p=\begin{cases} 0 & \text{ ~ \text{if}~ ~ }x=0\\ p^{-m}& \text{ if $p^ m$ is the factor with base $p$ in the decomposition~ ~ \ref{primi}\ .} \end{cases} \]

Finally, we define $d(x,y)=|x-y|_ p$, which will turn out to be a distance on $ℚ$, called $p$–adic distance.

We add this definition, which will be very useful in the following.

Definition 311

[0XG]For $n∈ℤ,n≠ 0$ we define

\[ 𝜑_ p(n)=\max \{ h∈ℕ, p^ h \text{ {divides} } n\} ~ ~ . \]

Let’s also define $𝜑_ p(0)=∞$. This $𝜑_ p$ is known as p-adic valuation [ 63 ] . .

E311

[0XH] Prove these fundamental relation.

$|1|_ p=1$ and more generally $|n|_ p≤ 1$ for every nonnull integer $n$, with equality if $n$ is not divisible by $p$.
Given $n$ nonnull integer, we have that $|n|_ p=p^{-𝜑_ p(n)}$.
Given $n,m$ integers, we have that $𝜑_ p(n+m)≥ \min \{ 𝜑_ p(n),𝜑_ p(m)\} $ with equality if $𝜑_ p(n)≠ 𝜑_ p(m)$.
Given $n,m$ nonzero integers, we have that $𝜑_ p(nm)=𝜑_ p(n)+𝜑_ p(m)$ and therefore $|nm|_ p=|n|_ p |m|_ p$.
Given $x=a/b$ with $a,b$ nonnull integers we have that $|x|_ p=p^{-𝜑_ p(a)+𝜑_ p(b)}$. Note that if $a,b$ are coprime, then one of the two terms $𝜑_ p(a),𝜑_ p(b)$ is zero.
Prove that $|x y|_ p = |x|_ p |y|_ p$ for $x,y∈ℚ$.
Prove that $|x/y|_ p = |x|_ p / |y|_ p$ for $x,y∈ℚ$ nonzero.

[UNACCESSIBLE UUID ’0XJ’]

[UNACCESSIBLE UUID ’0XK’] [0XM] Check that

\begin{equation} |x+y|_ p≤\max \big\{ |x|_ p,|y|_ p\big\} \label{eq:valu_ p-adi_ plus} \end{equation}

312

for each $x,\, y∈ℚ$. and therefore

\[ d_ p(x,z)≤\max \big\{ d_ p(x,y),d_ p(y,z)\big\} \ ,\qquad ∀\, x,\, y,\, z∈ ℚ\ ~ . \]

that is, this is an ultrametric (and therefore a distance). Hidden solution: [UNACCESSIBLE UUID ’0XN’] The properties 6 and ?? say that the p-adic valuation is an absolute value, and indeed it is a Krull valuation.

[UNACCESSIBLE UUID ’0XP’] [0XQ]Show that the multiplication map is continuous. Hidden solution: [UNACCESSIBLE UUID ’0XR’]

[UNACCESSIBLE UUID ’0XS’] [0XT] Find an example of a sequence that tends to zero (but never takes the value 0). This example shows that the associated topology is not the discrete topology. Hidden solution: [UNACCESSIBLE UUID ’0XV’] [0XW]Difficulty:*.Show, for every $a/b∈ℚ$ with $a,b$ coprime and $b$ not divisible by $p$, there exists $(x_ n)_ n⊆ ℤ$ such that $|x_ n-a/b|_ p→_ n 0$. Note that the assumption is necessary.

Hidden solution: [UNACCESSIBLE UUID ’0XX’] We proved that $ℤ$ is dense in the disk $\{ x∈ℚ, |x|_ p≤ 1\} $. [0XY]Difficulty:**.Show that $(ℚ,d)$ is not a complete metric space.

Hidden solution: [UNACCESSIBLE UUID ’0XZ’] [0Y0]Show that no $p$–adic distance on $ℚ$ is bi–Lipschitz equivalent to the Euclidean distance (induced by $ℝ$).

Hidden solution: [UNACCESSIBLE UUID ’0Y1’]

[UNACCESSIBLE UUID ’0Y2’]