Putting topologies on dp-minimal fields

4. If b ∈ acl(aS), then dp-rk(b/S) ≤ dp-rk(a/S). 5. dp-rk(ab/S) ≤ dp-rk(a/bS) + dp-rk(b/S). We work in a monster model C of a dp-minimal field (i.e., a field of dp-rank 1). “Models” will refer to small elementary substructures M C. We’ll write concatenation multiplicatively and multiplication using ·. We will assume that C is not strongly minimal. One already knows what strongly minimal fields look like, by Macintyre’s theorem—they are algebraically closed. If X and Y are subsets of U, we’ll use X + Y and X − Y to denote the usual things: X + Y = {x+ y : x ∈ X, y ∈ Y } X − Y = {x− y : x ∈ X, y ∈ Y }