## 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.

**What is p-adic expansion?**

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.

### What are p-adic groups?

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.

**Are p-adic numbers ordered?**

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.

**Are P Adics algebraically closed?**

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.

**Why are p-ADIC numbers interesting?**

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.