In this chapter p is always a (fixed) prime.
The p-adic numbers Qp are the completion of the rational numbers with respect to the valuation np(pv[(a)/(b)]) = v if p divides neither a nor b. They form a field of characteristic 0 which nevertheless shows some behaviour of the finite field with p elements.
A p-adic numbers can be approximated by a `` p-adic expansion'' which is similar to the decimal expansion used for the reals (but written from left to right). So for example if p = 2, the numbers 1, 2, 3, 4, [1/2] and [4/5] are represented as 1(2), 0.1(2), 1.1(2), 0.01(2), 10(2) and 0.0101(2). Approximation means to ignore powers of p, so for example with only 2 digits accuracy [4/5] would be approximated as 0.01(2). The important difference to the decimal approximation is that p-adic approximation is a ring homomorphism on the subrings of p-adic numbers whose valuation is bounded from below.
In GAP, p-adic numbers are represented by approximations. A family of (approximated) p-adic numbers consists of p-adic numbers with a certain precision and arithmetic with these numbers is done with this precision.
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GAP 4 manual