EDB — 0XM

Exercises

1. [0XM] Check that

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

   146

   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).

   Solution 1

   [0XN]↺↻

   The properties [E9.146f]↺↻ and ?? say that the p-adic valuation is an absolute value, and indeed it is a Krull valuation.

   [ [0XP]↺↻]