The reals and the p-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general algebraic number fields, in an analogous way. The remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers.

## Is the P-ADIC integers complete?

The p-adic integers can also be seen as the completion of the integers with respect to a p-adic metric. Let us introduce a p-adic valuation on the integers, which we will extend to Zp.

mathematician Kurt Hensel
Abstract. The p-adic numbers were invented at the beginning of the twentieth century by the German mathematician Kurt Hensel (1861–1941). The aim was to make the methods of power series expansions, which play such a dominant role in the theory of functions, available to the theory of numbers as well.

### How do you calculate P-ADIC expansion?

The proof of Theorem 3.1 gives an algorithm to compute the p-adic expansion of any rational number in Zp: (1) Assume r < 0. (If r > 0, apply the rest of the algorithm to −r and then negate with (2.2) to get the expansion for r.) (2) If r ∈ Z<0 then write r = −R and pick j ≥ 1 such that R < pj.

### How do you calculate P ADIC expansion?

Filters. (1) See AIDC. 1. (mathematics computing) When combined with prefixes derived (usually) from Latin or Greek names for numbers, used to make adjectives meaning “having a certain number of arguments” (said of functions, relations, etc, in mathematics and functions, operators, etc, in computing).

#### How is the p-adic expansion of a rational number defined?

The p – adic expansion of a rational number is defined similarly, but with a different division step. More precisely, given a fixed prime number p, every nonzero rational number where k is a (possibly negative) integer, and n and d are coprime integers both coprime with p. The integer k is the p-adic valuation of r, denoted

#### How is the metric space of the p-adic number complete?

This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure that gives the p -adic number systems their power and utility.

Can a field of fractions be written as a p adic number?

The ring of p-adic integers has no zero divisors, so we can take the field of fractions to get the field Q p of p-adic numbers. Note that in this field of fractions, every non-integer p-adic number can be uniquely written as p −n u with a natural number n and a unit u in the p-adic integers.

## Which is the finite representation of a p-adic number?

If a p -adic representation is finite on the left (that is, for large values of i), then it is the p -adic representation of a nonnegative rational number of the form with n an integer and These rational numbers are exactly the nonnegative rational numbers that have a finite representation in base p.