# 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.

Why are p ADIC numbers interesting?

The p-adics come up in homotopy theory. The main reason is because of their usefulness in the theory of formal group laws. They are also relevant in certain parts of algebraic geometry, they are (one of) the first examples of completions.

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 a field?

There is a unique field homomorphism from the rational numbers into the p-adic numbers, which maps a rational number to its p-adic expansion. This allows considering the p-adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the p-adic 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).

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.

## Is Q_P algebraic over Q?

Pi is transcendental over Q but algebraic over the field of real numbers R: it is the root of g(x) = x − π, whose coefficients (1 and −π) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses C/Q, not C/R.)

Is QA a field?

Q is a prime field, which is a field that has no subfield other than itself. The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to Q. Q is the field of fractions of the integers Z.

Is Q 2 a field?

and ·, but Q[ √ 2] is a field (a subfield of R). Q[ √ 2] is closed under addition. Inverses for addition: The inverse of a + b √ 2 is −(a + b √ 2) = −a + −b √ 2 ∈ Q[ √ 2].