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**Proposition 22**.: _Any triangle in an ultrametric space is isosceles and the length of its base does not exceed the length of sides._
Proof
: Suppose \(a\) and \(x\) are elements of a non-Archimedian field \(F\), satisfying \(||x-a||<||a||\), then using the non-Archimedian property
\[||x||=||x-a+a||\leq\max\{||x-a||,||a||\}=||a||.\]
On the other hand
\[||a||=||a-x+x||\leq\max\{||x-a||,||x||\}.\]
Now if \(||x-a||>||a||\), then by the above inequality we get \(||a||\leq||x-a||\) contrary to the assumption, thus
\[||a||=||a-x+x||\leq\max\{||x-a||,||x||\}\leq||x||.\]
So, \(||x||=||a||\). What we showed can be restated as follows: let \(a,b\) be elements in an ultrametric space, then
\[||b||<||a||\quad\text{implies}\quad||a+b||=||a||.\]
In other words, the longer side “wins” or if \(a,b\), and \(a+b\) are the sides of a triangle in an ultrametric space then this triangle is isosceles and the length of its base does not exceed the length of the sides.
### \(p\)-adic Absolute Value Metric
As mentioned above on the field \(\mathbb{Q}\) we have the Euclidean metric to measure the distance between two rational numbers. It is natural to wonder if the Euclidean distance is the most useful one. Is there any other way one can describe the closeness between rational numbers? It turns out that the \(p\)-adic absolute value function measures distance between rational numbers from the following construction.
**Definition 49**.: Let \(p\in\mathbb{Z}\) be a prime number. For any nonzero integer \(a\)_p-adic ordinal_ of \(a\) is denoted by \(\operatorname{ord}(a)\) and defined as
\[\operatorname{ord}_{p}(a)=\text{the largest power of $p$ which divides $a$}.\]
If \(x=\dfrac{a}{b}\) where \(a,b\in\mathbb{Z}\), \(b\neq 0\) then
\[\operatorname{ord}_{p}(x)=\operatorname{ord}_{p}(a)-\operatorname{ord}_{p}(b).\]
**Example 63**.: The following are simple examples for the concept of \(p\)-adic ordinals.
* \(\operatorname{ord}_{5}(22)=0\) since \(5\nmid 22\).
* \(\operatorname{ord}_{3}(423)=2\) since \(423=3^{2}\cdot 47\) and \(3\nmid 47\).
* \(\operatorname{ord}_{2}\left(\frac{4}{7}\right)=\operatorname{ord}_{2}(2^{2})- \operatorname{ord}_{2}(7)=2-0=2\).
**Definition 50**.: Let \(p\in\mathbb{Z}\) be a prime number. For any rational number \(x\), define a map \(|\cdot|_{p}\) on \(\mathbb{Q}\) called the _p-adic absolute value_ as follows:
\[|x|_{p}=\left\{\begin{array}{ll}p^{-\operatorname{ord}_{p}(x)}&\text{ if }\,x\neq 0\\ 0&\text{ if }\,x=0.\end{array}\right.\]
Going back to the above example \(|22|_{5}=5^{0}=1\), \(|423|_{3}=3^{-2}=\frac{1}{9}\) and \(\left(\frac{4}{7}\right)|_{2}=2^{-2}=\frac{1}{4}\).
**Proposition 23**.: \(|\cdot|_{p}\) _is a non-Archimedian norm on \(\mathbb{Q}\). The \(p\)-adic absolute value \(|\cdot|_{p}\) gives a metric on \(\mathbb{Q}\) defined by_
\[d(x,y)=|x-y|_{p}\quad x,y\in\mathbb{Q}.\]
Proof
: We need to show
* \(|x|_{p}\geq 0\) and \(|x|_{p}=0\) if and only if \(x=0\).
* \(|xy|_{p}=|x|_{p}|y|_{p}\).
* \(|x+y|_{p}\leq\max\{|x|_{p},|y|_{p}\}\).
a) is clear from the definition and b) follows from
\[\operatorname{ord}_{p}(xy)=\operatorname{ord}_{p}(x)+\operatorname{ord}_{p}(y).\]
It remains to show part c) holds. Note that if \(x=0\) or \(y=0\), then c) is true, so without loss of generality assume \(x,y\neq 0\). Let \(x=\frac{a}{b}\) and \(y=\frac{c}{d}\), then we have \(x+y=\frac{ad+bc}{bd}\) and
\[\operatorname{ord}_{p}(x+y)=\operatorname{ord}_{p}\left(\frac{ad+bc}{bd} \right)=\operatorname{ord}_{p}(ad+bc)-\operatorname{ord}_{p}(bd).\]
Now observe that the highest power of \(p\) dividing the sum \(ad+bc\) is greater than or equal to both the power of \(p\) that divides \(ad\) and the power of \(p\) that divides \(bc\). Therefore,
\[\operatorname{ord}_{p}(ad+bc)\geq\min\{\operatorname{ord}_{p}(ad), \operatorname{ord}_{p}(bc)\}.\]Furthermore, the highest power \(p\) that divides the product \(bd\) is just the sum of the power of \(p\) dividing \(b\) and dividing \(d\), i.e., \(\operatorname{ord}_{p}(bd)=\operatorname{ord}_{p}(b)+\operatorname{ord}_{p}(d)\). Combining this information we obtain
\[\operatorname{ord}_{p}(x+y)\geq\min\{\operatorname{ord}_{p}(ad),\operatorname{ ord}_{p}(bc)\}-\operatorname{ord}_{p}(b)-\operatorname{ord}_{p}(d)\]
or equivalently
\[\operatorname{ord}_{p}(x+y)\geq\min\{\operatorname{ord}_{p}(x),\operatorname{ ord}_{p}(y)\}.\]
It follows that \(-\operatorname{ord}_{p}(x+y)\geq\max\{\operatorname{ord}_{p}(x),\operatorname{ ord}_{p}(y)\}\). Therefore,
\[|x+y|_{p}=p^{-\operatorname{ord}_{p}(x+y)}\leq\max\{p^{-\operatorname{ord}_{p}( x)},p^{-\operatorname{ord}_{p}(y)}\}=\max\{|x|_{p},|y|_{p}\}.\]
Note that \(|\cdot|_{p}\) also satisfies the strong triangle inequality
\[|x+y|_{p}\leq\max\{|x|_{p},|y|_{p}\}\leq|x|_{p}+|y|_{p}.\qed\]
The \(p\)-adic metric plays a significant role in analysis as well as in number theory. \(\mathbb{Q}\) admits the \(p\)-adic norm \(|\cdot|_{p}\) for each prime \(p\) as well as the absolute value \(|\cdot|\). The following theorem, due to Ostrowski, states that this list is complete in some sense.
**Theorem 59** (Ostrowski’s Theorem).: _Every nontrivial norm \(||\cdot||\) on \(\mathbb{Q}\) is either equivalent to \(|\cdot|_{p}\) for some \(p\) or equivalent to \(|\cdot|\)._
For the proof of Ostrowski’s theorem we refer to, e.g., [30], pp. 43-45. One can now define _Cauchy sequences with respect to \(|\cdot|_{p}\)_ in the usual way.A sequence of rational numbers \(\{r_{i}\}\) is Cauchy in \(|\cdot|_{p}\) if for every \(\epsilon>0\), there is an integer \(N\) such that
\[|r_{i}-r_{j}|_{p}<\epsilon\quad\text{whenever }i,j>N.\]
It turns out that there is a Cauchy sequence in \(\mathbb{Q}\) with respect to \(|\cdot|_{p}\) that does not converge in \(\mathbb{Q}\) (consult [30], pp. 43-45 for a detailed discussion or for an example, set \(x_{n}=1+p+\cdots+p^{n}\), then show \(\{x_{n}\}\) is Cauchy in \((\mathbb{Q},|\cdot|_{p})\) and does not converge to any \(x\in\mathbb{Q}\)). This shows that \(\mathbb{Q}\) is not complete with respect to \(|\cdot|_{p}\).
**Definition 51**.: Let \(p\) be a fixed prime. _The field of \(p\)-adic numbers_ is denoted by \(\mathbb{Q}_{p}\) and defined to be the completion of \(\mathbb{Q}\) with respect to \(p\)-adic norm \(|\cdot|_{p}\).
Completing \(\mathbb{Q}\) to the real numbers \(\mathbb{R}\) is carried out by Cauchy sequences or by Dedekind cuts; similarly we think of elements of \(\mathbb{Q}_{p}\) as equivalence classes of Cauchy sequences from \(\mathbb{Q}\) with respect to \(|\cdot|_{p}\). We can define addition and multiplication on \(\mathbb{Q}_{p}\) so that it becomes a field. To learn more about algebraic properties of \(\mathbb{Q}_{p}\), one can consult, e.g., [43], pp. 176-184.
### The Topology of \(\mathbb{R}\) vs. the Topology of \(\mathbb{Q}_{p}\)
In this section we compare \(\mathbb{R}\) and \(\mathbb{Q}_{p}\) from the point of view of metric spaces. In \(\mathbb{R}\) an open ball \(B_{r}(a)\) with radius \(r\) centered at \(a\) is defined to be
\[B_{r}(a):=\{x\in\mathbb{R}:\ |x-a| In \(\mathbb{Q}_{p}\) open balls are the sets \[B_{r}(a)_{p}:=\{x\in\mathbb{Q}_{p}:\ |x-a|_{p}
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