Habeas corpus

In Mathematical, a number p - adic is an element of the body $\backslash \; mathbb\; Q\_p$ of the numbers p - adic, where p is a Prime number given. One thus speaks about dyadic number , triadic , etc

The bodies $\backslash \; mathbb\; Q\_p$ of the numbers p - adic are built by completion body $\backslash \; mathbb\; Q$ of the rational numbers when this one is provided with a named particular standard standard p - adic and noted $|.\; |\_p$. In a sense, the bodies $\backslash \; mathbb\; Q\_p$ are related with the body $\backslash \; R$ of the real numbers, which is also a completion of the body of the rational numbers when the standard considered is the usual absolute value.

The main motivation having given rise to the bodies of the numbers p - adic was to be able to use the techniques of the whole series in the Théorie of the numbers, but their utility now largely exceeds this framework. Moreover, the standard p - adic on the body $\backslash \; mathbb\; Q\_p$ is a standard non-archimédienne: one obtains on this body an analysis different from the analyzes usual on realities, which one calls p-adic Analyze.

Construction

Analytical approach

The real numbers are defined like classes of equivalence of the series Cauchy of the rational numbers. However, this definition rests on the Métrique chosen and, while choosing some another, other numbers which the real numbers can be built. The metric one used for the real numbers is called metric Euclidean.

For a Prime number given $p$, one as follows defines the standard p - adic on $\backslash \; mathbb\; Q$:

one recalls that the valuation p - adic of an entirety not no one has is the exhibitor of p in the decomposition of has in product of factors first.

one can then build a valuation for any rational number not no one while posing:

$v\_p\; \backslash \; left\; (\backslash \; frac\; ab\; \backslash \; right)\; =\; v\_p\; (a)\; -\; v\_p\; (b)$.

One proves easily that this definition is independent of the representative of rational selected.

the Standard p - adic $|R|\_p$ of rational a $r$ not no one is worth $p^\; \{-\; v\_p\; (R)\}$.

If R is null, one poses $|R|\_p\; =\; 0$. This prolongation is compatible with the idea that 0 are divisible by $p^k$ for any value of K, therefore that the valuation of 0 would be infinite.

To some extent, more $r$ is divisible by $p$, more its standard p - adic is small (it is a particular case of discrete valuation an algebraic tool!).

For example, for $r\; =\; \{63\; \backslash \; over\; 550\}\; =\; 2^\; \{-\; 1\}\; \backslash \; times\; 3^2\; \backslash \; times\; 5^\; \{-\; 2\}\; \backslash \; times\; 7\; \backslash \; times\; 11^\; \{-\; 1\}$:

$|R|\_2=2\; \backslash ,$

$|R|\_3=\; \{1\; \backslash \; over\; 9\}\; \backslash ,$

$|R|\_5=25\; \backslash ,$

$|R|\_7=\; \{1\; \backslash \; over\; 7\}\; \backslash ,$

$|R|\_\; \{11\}\; =11\; \backslash ,$

$|R|\_p=1\; \backslash ,$ for any other prime number.

It is shown that this application has all the properties of a standard. One can show that any standard (not-commonplace) on $\backslash \; mathbb\; Q$ is equivalent either to the euclidian norm, or to a standard p - adic (Théorème of Ostrowski). A standard p - adic metric a $d\_p$ defines on $\backslash \; mathbb\; Q$ while posing:

$d\_p\; (X,\; there)\; =|X\; there|\_p$

The body $\backslash \; mathbb\; Q\_p$ of the numbers p - adic can then be defined like the completion of metric space ($\backslash \; mathbb\; Q$, $d\_p$). Its elements are the classes of equivalences of the series Cauchy, where two continuations are known as equivalent if their difference converges towards zero. In this way, one obtains a complete metric space which is also a body and which contains $\backslash \; mathbb\; Q$.

This construction makes it possible to include/understand why $\backslash \; mathbb\; Q\_p$ is an arithmetic analog of $\backslash \; mathbb\; R$.

Algebraic approach

In this algebraic approach, one starts by defining the ring entireties p - adic, then by construction the Corps of the fractions of this ring to obtain the body of the numbers p - adic.

One defines the ring of the entireties p - adic $\backslash \; mathbb\; Z\_p$ like the projective Limite of the rings $\backslash \; mathbb\; Z/p^n\; \backslash \; mathbb\; Z$. An entirety p - adic is then a continuation $(a\_n)\; \_\; \{N\; \backslash \; Ge\; 1\}$ such as $a\_n\; \backslash \; in\; \backslash \; mathbb\; Z/p^n\; \backslash \; mathbb\; Z$ and that, if

For example, $35$ as a number 2-adic would be the continuation $(1,\; 3,3,3,3,35,35,35\; \backslash \; ldots)$.

The addition and the multiplication of such continuations are well defined, since they commutate with the operator modulo (see modular Arithmétique). Moreover, any continuation $(a\_n)$ whose first element is not null has a reverse.

The ring of the entireties p - adic not having dividing of zero, it is possible to consider its Corps of the fractions to obtain the body $\backslash \; mathbb\; Q\_p$ of the numbers p - adic.

Canonical decomposition of Hensel

That is to say $p$ a prime number. Any element not no one $r$ of $\backslash \; mathbb\; Q\_p$ (and in particular any element of $\backslash \; mathbb\; Q$) is written in a single way in the form:

$r\; =\; \backslash \; sum\_\; \{i=k\}\; ^\; \backslash \; infty\; a\_i\; p^i$

where $k\; \backslash \; in\; \backslash \; Z$ and the $a\_i$ are integers ranging between $0$ and $p-1$. This writing is the canonical decomposition of $r$ like number p - adic.

This series is convergent according to metric the p - adic.

One notes $\backslash \; Z\_p$ all the elements of $\backslash \; mathbb\; Q\_p$ such as $k\; \backslash \; Ge\; 0$ and one calls together it entireties p - adic. $\backslash \; Z\_p$ is a subring of $\backslash \; mathbb\; Q\_p$. One can represent an entirety p - adic by an infinite continuation towards the left of figures in p bases, while the other elements of $\backslash \; mathbb\; Q\_p$, them, will have a finished number of figures on the right of the comma. This writing functions all in all contrary to what one with the practice to meet in the writing of the real numbers.

For example, with $p\; =\; 2$:

• $1\; =\; 1\; \backslash \; times\; 2^0\; =\; \backslash \; ldots\; 000001\_2$ (the 2 in index indicating that it is about the development 2-adic of 1)

• $-1\; =\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; 2^n\; =\; \backslash \; ldots\; 11111111111111\_2$: one can check that, since $\backslash \; ldots\; 001\_2+\; \backslash \; ldots\; 001\_2=\; \backslash \; ldots\; 0010\_2$, to add 1 to this writing results in shifting a reserve all along the writing, for finally giving 0.
• $3\; =\; \backslash \; ldots\; 000011\_2$
• $\{1\; \backslash \; over\; 3\}\; =\; 1\; +\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; 2^\; \{2n+1\}\; =\; \backslash \; ldots\; 01010101011\_2$: by multiplying this result by $\backslash \; ldots\; 000011\_2$, one finds 1.
• $\backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; 2^\; \{2^n\}$ represents an element of $\backslash \; mathbb\; Q\_p$ (and even of $\backslash \; mathbb\; Z\_p$) which is not in $\backslash \; mathbb\; Z$.

Another example, with $p\; =\; 7$:

2 does not have a square root in $\backslash \; mathbb\; Q$ but has of them one in $\backslash \; mathbb\; Q\_7$, namely $\backslash \; sqrt\; \{2\}\; =\dots \; 16244246442640361054365536623164112011266421216213\_7$.

Properties

Denombrability

The whole of the entireties p - adic countable is not .

The numbers p - adic contain the rational numbers and form a characteristic body of null. It is not possible to make of it a Corps ordered.

Topology

The Topology on all the entireties p - adic is that of the Ensemble of Cantor; topology on the whole of the numbers p - adic is that of the whole of private Cantor of a point (which would be naturally called infinite). In particular, the space of the entireties p - adic compact is , while the space of the numbers p - adic is to it only locally. As a metric spaces, entireties and the numbers p - adic are complete.

The real numbers have only one algebraic Extension clean, the complex numbers. In other words, this quadratic extension is algebraically closed. The algebraic end of the numbers p - adic is infinite. The bodies $\backslash \; mathbb\; Q\_p$ have an infinity of nonequivalent algebraic extensions. Moreover, the algebraic fence of a $\backslash \; mathbb\; Q\_p$ is not complete. Its metric completion is called $\backslash \; Omega^p$ and it is algebraically closed.

The body $\backslash \; Omega^p$, also noted $\backslash \; mathbb\; C\_p$, is abstractedly isomorphous with the body $\backslash \; mathbb\; C$ of the complex numbers and it is possible to regard the first as the last, provided with a metric exotic. It should however be noted that the existence of such an isomorphism is a consequence of the Axiome of the choice and that it is not possible to clarify one of them.

The numbers p - adic contain to it cyclotomic Corps if and only if $n$ divides $p-1$. For example, 1st, 2nd, 3rd, 4th, 6th and 12th cyclotomic bodies are subfields of $\backslash \; mathbb\; Q\_\; \{13\}$.

The number E is not element of any the bodies p - adic. However, $e^p$ is a number p - adic, except if $p=2$. $e$ is an element of the algebraic fence of all the bodies p - adic.

On the real numbers, only the functions whose Dérivée S are null is the constant functions. This is not true on the numbers p - adic. For example, the function

have a null derivative in all points, but is not even constant locally into 0.

If the elements $r\; are\; given,\; r\_\; 2,\; r\; \_\; 3,\; r\; \_\; 5,\; r\; \_7\; \backslash \; ldots$ respectively members of $\backslash \; R,\; \backslash \; mathbb\; Q\_2,\; \backslash \; mathbb\; Q\_3,\; \backslash \; mathbb\; Q\_5,\; \backslash \; mathbb\; Q\_7\; \backslash \; ldots$, it is possible to find a continuation $(x\_n)$ of $\backslash \; mathbb\; Q$ such as the limit of the $x\_n$ in $\backslash \; R$ either $r$ and, for all $p$ first, it is $r\_p$ in $\backslash \; mathbb\; Q\_p$.

Rationality

A positive number $\backslash \; gamma\_0$ is rational if, and only if, its development p-adic is periodic starting from a certain row, i.e., if there exist 2 entireties $N\; \backslash \; geq\; 0$ and $k\; >\; 0$ such as $\backslash \; forall\; N\; \backslash \; geq\; NR,\; a\_\; \{n+k\}\; =a\_\; \{K\}$ (the continuation $a\_n$ representing the development p-adic of the number $\backslash \; gamma\_0$)

Internal bonds

• Development in series of Engel

• Produces infinite of Cantor

Zh-classical: 進數

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