When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q [member of] C, or p-adic number q [member of] [C.
Rim, Generalized Carlitz's q-Bernoulli numbers in the p-adic number field, Adv.
11 For v [greater than or equal to] -1, the p-adic numbers
Other topics are rational points on elliptic curves, conics and the p-adic numbers
, the zeta function, and algebraic number theory.
In 1897, Hensel (5) discovered the p-adic numbers
as a number theoretical analogue of power series in complex analysis.
Using mainly concrete constructions, Gerstein gives a brief introduction to classical forms, then moves to quadratic spaces and lattices, valuations, local fields, p-adic numbers
, quadratic spaces over Qp and over Q, lattices over principal ideal domains, initial integral results, the local-global approach to lattices, and applications to cryptography.
They conclude by explaining the field of p-adic numbers
, their squares, absolute values and valuations, the topologies of valuation type, local fields and locally compact fields.