Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields Jan E. Holly 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 d (x , z ) ≤ max{d (x , y ), d ( y , z )} for all x , y , z ∈ X . In such a space, various interesting things happen: Triangles are always isoceles with the unequal side (if any) being shortest; every point in a given disc is a center of that disc; two discs can intersect only by having one completely contained in the other. If we imagine an ultrametric space as having its points on a line or in a plane, we cannot appeal to our usual intuition for distance. Instead, it is useful to have a new framework for visualizing the ultrametric space, and we propose using a different picture—that of a tree. One well-known ultrametric space is Q p , the field of p-adic numbers, and it is our main example in Section 2. The tree picture serves for other valued fields as well, and we discuss this in Section 4, after looking more closely at triangles and discs in Section 3. Before embarking on a study of these particular fields, however, it is instructive to discuss another example. Consider the set of all species in the animal kingdom, and look at the tree of classi- fication for these species in Figure 1. phylum subphylum class order family genus species Figure 1. October 2001] ULTRAMETRIC SPACES, p-ADIC NUMBERS, VALUED FIELDS 721