Height Functions

Definition 1. An (archimedean) absolute value on a field k is a real valued function ‖ · ‖ : k → [0,∞) with the following three properties: (1) ‖x‖ = 0 if and only if x = 0. (2) ‖xy‖ = ‖x‖ · ‖y‖. (3) ‖x+ y‖ ≤ ‖x‖+ ‖y‖. A nonarchimedean absolute value satisfies the extra condition that (3’) ‖x+ y‖ ≤ max{‖x‖, ‖y‖}. If k is a field, we denote Mk the set of absolute values on k. As an abuse of notation we say that if | · |v is an absolute value on k then v ∈ Mk rather then | · |v ∈Mk. The rational numbers Q have the archimedean absolute value |x|∞ = max{x,−x}. For each prime number p ∈ Z there is a nonarchimedean absolute value (usually called the p-adic absolute value) |x|p = p−ordp(x). Where ordp(x) is the unique integer such that x can be written in the form x = pp · a/b with a, b ∈ Z and p ab . Note 1. It can be proved that up to a ”power” that these are the only nontrivial absolute values on Q. See [Lang Number Theory] or [B-S Number Theory] Theorem 2. (Power Rule 1) ∏