Valuation ( $v$ )

A valuation assigns sizes to elements of a field in a way compatible with multiplication and addition. In additive notation,

v (x y) = v (x) + v (y)

and

v (x + y) \geq min {v (x), v (y)}

.

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- Non-Archimedean valuation Valuation

Non-Archimedean valuation

A non-Archimedean valuation satisfies the strong triangle inequality

v (x + y) \geq min {v (x), v (y)} .

In multiplicative absolute-value notation this becomes

∣ x + y ∣ \leq max {∣ x ∣, ∣ y ∣}

.

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