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[parent] ultrametric triangle inequality (Theorem)
Theorem 1   Let $ K$ be a field and $ G$ an ordered group equipped with zero. Suppose that the function $ \vert\cdot\vert: K\to G$ satisfies the postulates 1 and 2 of Krull valuation. Then the non-archimedean or ultrametric triangle inequality

3.                  $ \vert x+y\vert\leqq\max\{\vert x\vert,\,\vert y\vert\}$

in the field is equivalent with the condition

(*) $ \quad\quad\quad \vert x\vert\leqq 1 \,\,\,\, \Rightarrow \,\,\,\, \vert x+1\vert\leqq 1.$

Proof. The value $ y = 1$ in the ultrametric triangle inequality gives the (*) as result. Secondly, let's assume the condition (*). Let $ x$ and $ y$ be non-zero elements of the field $ K$ (if $ xy =0$ then 3 is at once verified), and let e.g. $ \vert x\vert \leqq \vert y\vert$. Then we get $ \displaystyle\vert\frac{x}{y}\vert = \vert x\vert\cdot\vert y\vert^{-1}\leqq 1$, and thus according to (*),

$\displaystyle \vert x+y\vert\cdot\vert y\vert^{-1} = \left\vert\frac{x+y}{y}\right\vert = \left\vert\frac{x}{y}+1\right\vert\leqq 1.$
So we see that $ \vert x+y\vert\leqq \vert y\vert = \max\{\vert x\vert,\,\vert y\vert\}$.
Theorem 2   The Krull valuation (and any non-archimedean valuation) $ \vert\cdot\vert$ of the field $ K$ satisfies the sharpening
$\displaystyle \vert x+y\vert = \max\{\vert x\vert,\,\vert y\vert\}\quad\mathrm{for}\,\,\,\vert x\vert \neq \vert y\vert$
of the ultrametric triangle inequality.

Proof. Let e.g. $ \vert x\vert > \vert y\vert$. Surely $ \vert x+y\vert \leqq \vert x\vert$, but also $ \vert x\vert = \vert(x+y)-y\vert \leqq \max\{\vert x+y\vert,\,\vert y\vert\}$; this maximum is $ \vert x+y\vert$ since otherwise one would have $ \vert x\vert \leqq \vert y\vert$. Thus the result is: $ \vert x+y\vert = \vert x\vert$.

Note. The metric defined by a non-archimedean valuation of the field $ K$ is the ultrametric of $ K$. Theorem 2 implies, that every triangle of $ K$ with vertices $ A$, $ B$, $ C$ ($ \in K$) is isosceles: if $ \vert B-C\vert \neq \vert C-A\vert$, then $ \vert A-B\vert = \max\{\vert B-C\vert,\,\vert C-A\vert\}$.

Theorem 3   The valuation $ \vert\cdot\vert: K\to \mathbb{R}$ of the field $ K$ is archimedean if and only if the set
$\displaystyle \{\vert 1\vert,\,\vert 1+1\vert,\,\vert 1+1+1\vert,\,\ldots\}$
of the “values” of the multiples of the unity is not bounded.

Proof. If $ \vert\cdot\vert$ is non-archimedean, then $ \vert n\cdot 1\vert = \vert 1+\ldots+1\vert \leqq\max\{\vert 1\vert\} = 1$, and the multiples are bounded. Conversely, let $ \vert n\cdot1\vert < M \,\, \forall n\in\mathbb{Z}_+$. Now one obtains, when $ \vert x\vert\leqq 1$:

$\displaystyle \vert x+1\vert^n \leqq \sum_{j = 0}^n \left\vert{n\choose j}\right\vert\cdot\vert x\vert^j < (n+1)M,$
or $ \vert x+1\vert < \sqrt[n]{(n+1)M}$ for all $ n$. As $ n$ tends to infinity, this $ n^\mathrm{th}$ root has the limit 1. Therefore one gets the limit inequality $ \vert x+1\vert \leqq 1$, i.e. the valuation is non-archimedean.

Bibliography

1
EMIL ARTIN: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).



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See Also: maximal number, p-adic canonical form, ultrametric space, minimal and maximal number, exponent valuation

Also defines:  non-archimedean triangle inequality

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Cross-references: inequality, limit, root, bounded, unity, multiples, archimedean, isosceles, vertices, triangle, implies, ultrametric, valuation, metric, non-archimedean, Krull valuation, postulates, function, ordered group equipped with zero, field
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This is version 20 of ultrametric triangle inequality, born on 2004-12-16, modified 2008-06-22.
Object id is 6587, canonical name is UltrametricTriangleInequality.
Accessed 4294 times total.

Classification:
AMS MSC11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous)
 12J20 (Field theory and polynomials :: Topological fields :: General valuation theory)
 13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations)
 13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings)

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