mathbbQp

The notation mathbbQp refers to the field of p-adic rational numbers. It is constructed as the field of fractions of the ring of p-adic integers, denoted mathbbZp. The p-adic integers are a completion of the integers with respect to a specific metric, the p-adic metric, which is based on the prime number p. This completion results in a structure that shares some properties with the real numbers but also exhibits fundamentally different characteristics.

The p-adic rational numbers are formed by taking ratios of p-adic integers where the denominator is not divisible by p in the p-adic sense. This construction ensures that mathbbQp forms a field, meaning that every non-zero element has a multiplicative inverse. Unlike the real numbers, which are topologically complete with respect to the standard absolute value metric, mathbbQp is complete with respect to the p-adic absolute value. This difference leads to unique analytical properties. For instance, in mathbbQp, a convergent series is not necessarily a Cauchy sequence in the traditional sense. The field mathbbQp plays a crucial role in number theory, algebraic geometry, and p-adic analysis.