The p-adic valuation is a mathematical concept used to measure how divisible a number is by a prime number p. It assigns a non-negative integer to each integer, indicating the highest power of p that divides that integer. For example, if we take the number 12 and the prime 3, the 3-adic valuation of 12 is 1, since 3^1 (which is 3) divides 12, but 3^2 (which is 9) does not. This valuation is part of the broader field of p-adic numbers, which extends the usual number system to include these valuations. The p-adic numbers are useful in various areas of mathematics, including number theory and algebraic geometry, as they provide a different way to analyze numbers and their properties.