Pictures of Ultrametric Spaces, the p-Adic Numbers, and Valued Fields

1. INTRODUCTION. When studying a metric space, it is valuable to have a mental picture that displays distance accurately. When the space is Z, Q, or R, we usually form such a picture by imagining points on the " number line ". When the space is X = Z 2 , Q 2 , R 2 , or C we use a planar picture in which nonempty discs (sets of the form {x ∈ X : d(x, b) ≤ γ } or {x ∈ X : d(x, b) < β}, with metric d, point b ∈ X, γ nonnegative, and β positive) can be drawn on paper with a circular shape, and the triangle inequality is demonstrated by drawings of triangles. This assumes that the standard metric is in use, but even with a slightly different metric, the planar picture might still be useful; discs might be diamond-shaped instead of round, for example. However, we find that if the space is non-Archimedean (i.e., if it is an ultrametric space), then the usual pictures lose their utility. An ultrametric space X is a metric space in which the metric satisfies the strong triangle inequality