Common questions

# What are P-ADIC numbers used for?

## What are P-ADIC numbers used for?

The p-adic absolute value gives us a new way to measure the distance between two numbers. The p-adic distance between two numbers x and y is the p-adic absolute value of the number x-y. So going back to the 3-adics, that means numbers are closer to each other if they differ by a large power of 3.

The p-adic expansion of a rational number is eventually periodic. Conversely, a series with. converges (for the p-adic absolute value) to a rational number if and only if it is eventually periodic; in this case, the series is the p-adic expansion of that rational number.

An analytic group over the field Qp of p-adic numbers (more generally, over a locally compact non-Archimedean field K). Natural examples of p-adic Lie groups are the Galois groups of certain infinite extensions of fields.

What does ADIC mean in math?

Filters. (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).

## Are P Adics a field?

The formally p-adic fields can be viewed as an analogue of the formally real fields. and its residue field has 9 elements. When F is formally p-adic but that there does not exist any proper algebraic formally p-adic extension of F, then F is said to be p-adically closed.

Here’s a final curious fact about the p-adic numbers. We all know that if x and y are two non-equal real numbers then either xthere is no linear ordering of the p-adic numbers!

### Are p-adic integers a field?

For p any prime number, the p-adic numbers (or p-adic rational numbers, for emphasis) form a field ℚp that completes the field of rational numbers with respect to a metric, called the p-adic metric. As such they are analogous to real numbers.

When F is formally p-adic but that there does not exist any proper algebraic formally p-adic extension of F, then F is said to be p-adically closed. For example, the field of p-adic numbers is p-adically closed, and so is the algebraic closure of the rationals inside it (the field of p-adic algebraic numbers).

## 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.

Since p-adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of p the higher the power the closer they are.

### Is there an introduction to p-adic analysis?

These notes have no pretension to being a thorough introduction to p-adic analysis. Such topics as the Hasse-Minkowski Theorem (which is in Chapter 1 of Borevich and Shafarevich’s Number Theory) and Tate’s thesis (which is also available in textbook form, see Lang’s Algebraic Number Theory) are omitted. IX

How is p-adic Numbers, Ultrametric Analysis and applications peer reviewed?

P-Adic Numbers, Ultrametric Analysis, and Applications is a peer reviewed journal. We use a single blind peer review format. Our team of reviewers includes 26 experts, both internal and external (70%), from 13 countries. The average period from submission to first decision in 2019 was 60 days, and that from first decision to acceptance was 23 days.

## How is the p-adic number system related to real numbers?

The 3-adic integers, with selected corresponding characters on their Pontryagin dual group In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

Who is the founder of the p adic system?

The p-adic numbers are an intuitive arithmetic system (but geometrically counterintuitive) that was discovered by German mathematician Kurt Hensel in about 1899 and by German mathematician Ernst Kummer (1810-1893) earlier in elementary form. The closely related adeles and ideles were introduced in the 1930s by Claude Chevalley and André Weil.