Absolute Value (algebra) - Valuations

Valuations

If for some ultrametric absolute value we define ν(x) = -log_b| x | for any base b > 1, and extend by defining ν(0) = ∞, which is ordered to be greater than all real numbers, we obtain a function from D to R ∪ {∞}, with the following properties:

• ν(x) = ∞ ⇒ x = 0,
• ν(xy) = ν(x) + ν(y),
• ν(x + y) ≥ min(ν(x), ν(y)).

Such a function is known as a valuation in the terminology of Bourbaki, but other authors use the term valuation for absolute value and then say exponential valuation instead of valuation.